Toward a Theory of Chaos

December 3, 2003 12:13 00851

Tutorials and Reviews

International Journal of Bifurcation and Chaos, Vol. 13, No. 11 (2003) 3147–3233 c World Scientiﬁc Publishing Company

TOWARD A THEORY OF CHAOS

A. SENGUPTA Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India [email protected]

Received February 23, 2001; Revised September 19, 2002

This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000], and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this uniﬁed mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain eﬃciently a multiplicity of states that otherwise would remain unavailable to it.

Keywords: Chaos; complexity; ill-posed problems; graphical convergence; topology; multifunctions.

Prologue of study of so-called “strongly” nonlinear system. Linearity means that the rule that determines what 1. Generally speaking, the analysis of chaos is ex- . . . tremely difficult. While a general definition for chaos a piece of a system is going to do next is not influ- applicable to most cases of interest is still lacking, enced by what it is doing now. More precisely this mathematicians agree that for the special case of iter- is intended in a differential or incremental sense: For ation of transformations there are three common char- a linear spring, the increase of its tension is propor- acteristics of chaos: tional to the increment whereby it is stretched, with the ratio of these increments exactly independent of 1. Sensitive dependence on initial conditions, how much it has already been stretched. Such a spring 2. Mixing, can be stretched arbitrarily far Accordingly no real 3. Dense periodic points. . . . . spring is linear. The mathematics of linear objects is [Peitgen, Jurgens & Saupe, 1992] particularly felicitous. As it happens, linear objects en- 2. The study of chaos is a part of a larger program joy an identical, simple geometry. The simplicity of

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this geometry always allows a relatively easy mental 5. One of the most striking aspects of physics image to capture the essence of a problem, with the is the simplicity of its laws. Maxwell’s equations, technicality, growing with the number of parts, basi- Schroedinger’s equations, and Hamilton mechanics cally a detail. The historical prejudice against nonlinear can each be expressed in a few lines. . . . Everything problems is that no so simple nor universal geometry is simple and neat except, of course, the world. Every usually exists. place we look outside the physics classroom we see a world of amazing complexity. So why if the laws Mitchell Feigenbaum’s Foreword (pp. 1–7) . . . , are so simple is the world so complicated To us com- in [Peitgen et al., 1992] , ? plexity means that we have structure with variations. 3. The objective of this symposium is to explore the Thus a living organism is complicated because it has impact of the emerging science of chaos on various dis- many different working parts, each formed by varia- ciplines and the broader implications for science and tions in the working out of the same genetic coding. society. The characteristic of chaos is its universality Chaos is also found very frequently. In a chaotic world and ubiquity. At this meeting, for example, we have it is hard to predict which variation will arise in a given scholars representing mathematics, physics, biology, place and time. A complex world is interesting because geophysics and geophysiology, astronomy, medicine, it is highly structured. A chaotic world is interesting psychology, meteorology, engineering, computer sci- because we do not know what is coming next. Our ence, economics and social sciences.1 Having so many world is both complex and chaotic. Nature can pro- disciplines meeting together, of course, involves the duce complex structures even in simple situations and risk that we might not always speak the same lan- obey simple laws even in complex situations. guage, even if all of us have come to talk about [Goldenfeld & Kadanoff, 1999] “chaos”. 6. Where chaos begins, classical science stops. For as Opening address of Heitor Gurgulino de Souza, long as the world has had physicists inquiring into Rector United Nations University, Tokyo the laws of nature, it has suffered a special ignorance [de Souza, 1997] about disorder in the atmosphere, in the turbulent sea, 4. The predominant approach (of how the different in the fluctuations in the wildlife populations, in the fields of science relate to one other) is reductionist: oscillations of the heart and the brain. But in the 1970s Questions in physical chemistry can be understood a few scientists began to find a way through disor- in terms of atomic physics, cell biology in terms of der. They were mathematicians, physicists, biologists, how biomolecules work . . . . We have the best of rea- chemists . . . (and) the insights that emerged led di- sons for taking this reductionist approach: it works. rectly into the natural world: the shapes of clouds, But shortfalls in reductionism are increasingly appar- the paths of lightning, the microscopic intertwining ent (and) there is something to be gained from sup- of blood vessels, the galactic clustering of stars. . . . plementing the predominantly reductionist approach Chaos breaks across the lines that separate scientific with an integrative agenda. This special section on disciplines, (and) has become a shorthand name for a complex systems is an initial scan (where) we have fast growing movement that is reshaping the fabric of taken a “complex system” to be one whose properties the scientific establishment. are not fully explained by an understanding of its com- [Gleick, 1987] ponent parts. Each Viewpoint author 2 was invited to define “complex” as it applied to his or her discipline. 7. order ( ) complexity ( ) chaos. → → [Gallagher & Appenzeller, 1999] [Waldrop, 1992]

1A partial listing of papers is as follows: Chaos and Politics: Application of Nonlinear Dynamics to Socio-Political issues; Chaos in Society: Reﬂections on the Impact of Chaos Theory on Sociology; Chaos in Neural Networks; The Impact of Chaos on Mathematics; The Impact of Chaos on Physics; The Impact of Chaos on Economic Theory; The Impact of Chaos on Engineering; The impact of Chaos on Biology; Dynamical Disease: And The Impact of Nonlinear Dynamics and Chaos on Cardiology and Medicine. 2The eight Viewpoint articles are titled: Simple Lessons from Complexity; Complexity in Chemistry; Complexity in Biolog- ical Signaling Systems; Complexity and the Nervous System; Complexity, Pattern, and Evolutionary Trade-Oﬀs in Animal Aggregation; Complexity in Natural Landform Patterns; Complexity and Climate, and Complexity and the Economy. December 3, 2003 12:13 00851

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8. Our conclusions based on these examples seem sim- essary that we have a mathematically clear physi- ple: At present chaos is a philosophical term, not a cal understanding of these notions that are suppos- rigorous mathematical term. It may be a subjective edly reshaping our view of nature. This paper is an notion illustrating the present day limitations of the attempt to contribute to this goal. To make this human intellect or it may describe an intrinsic prop- account essentially self-contained we include here, erty of nature such as the “randomness” of the se- as far as this is practical, the basics of the back- quence of prime numbers. Moreover, chaos may be ground material needed to understand the paper in undecidable in the sense of Godel in that no matter the form of Tutorials and an extended Appendix. what definition is given for chaos, there is some ex- The paradigm of chaos of the kneading of the ample of chaos which cannot be proven to be chaotic dough is considered to provide an intuitive basis from the definition. of the mathematics of chaos [Peitgen et al., 1992], [Brown & Chua, 1996] and one of our fundamental objectives here is to re- count the mathematical framework of this process 9. My personal feeling is that the definition of a “frac- in terms of the theory of ill-posed problems arising tal” should be regarded in the same way as the biolo- from non-injectivity [Sengupta, 1997], maximal ill- gist regards the definition of “life”. There is no hard posedness, and graphical convergence of functions and fast definition, but just a list of properties char- [Sengupta & Ray, 2000]. A natural mathematical acteristic of a living thing . . . . Most living things have formulation of the kneading of the dough in the most of the characteristics on the list, though there form of stretch-cut-and-paste and stretch-cut-and- are living objects that are exceptions to each of them. fold operations is in the ill-posed problem arising In the same way, it seems best to regard a fractal as from the increasing non-injectivity of the function a set that has properties such as those listed below, f modeling the kneading operation. rather than to look for a precise definition which will certainly exclude some interesting cases. [Falconer, 1990] Begin Tutorial 1: Functions and 10. Dynamical systems are often said to exhibit chaos Multifunctions without a precise definition of what this means. A relation, or correspondence, between two sets X [Robinson, 1999] and Y , written : X– Y , is basically a rule that M → associates subsets of X →to subsets of Y ; this is often 1. Introduction expressed as (A, B) where A X and B Y ∈ M ⊂ ⊂ The purpose of this paper is to present an unified, and (A, B) is an ordered pair of sets. The domain self-contained mathematical structure and physical def ( ) = A X : ( Z )(π (Z) = A) understanding of the nature of chaos in a discrete D M { ⊂ ∃ ∈ M X } dynamical system and to suggest a plausible expla- and range nation of why natural systems tend to be chaotic. def ( ) = B Y : ( Z )(π (Z) = B) The somewhat extensive quotations with which we R M { ⊂ ∃ ∈ M Y } begin above, bear testimony to both the increas- of are respectively the sets of X which under M ingly significant — and perhaps all-pervasive — corresponds to sets in Y ; here π and (π ) M X Y role of nonlinearity in the world today as also our are the projections of Z on X and Y , respectively. imperfect state of understanding of its manifesta- Equivalently, ( ( ) = x X : (x) = ) and D M { ∈ M 6 ∅} tions. The list of papers at both the UN Confer- ( ( ) = (x)). The inverse − of R M x∈D(M) M M M ence [de Souza, 1997] and in Science [Gallagher & is the relation Appenzeller, 1999] is noteworthy if only to justify S − = (B, A) : (A, B) the observation of Gleick [1987] that “chaos seems M { ∈ M} to be everywhere”. Even as everybody appears to so that − assigns A to B iff assigns B to A. M M be finding chaos and complexity in all likely and In general, a relation may assign many elements in unlikely places, and possibly because of it, it is nec- its range to a single element from its domain; of December 3, 2003 12:13 00851

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especial significance are functional relations f 3 that linear homogeneous differential equation with con- can assign only a unique element in (f) to any stant coefficients of order n > 1 has n linearly element in (f). Figure 1 illustratesRthe distinc- independent solutions so that the operator Dn of tion betweenDarbitrary and functional relations Dn(y) = 0 has a n-dimensional null space. Inverses and f. This difference between functions (or maps)M of non-injective, and in general non-bijective, func- − and multifunctions is basic to our development and tions will be denoted by f . If f is not injective should be fully understood. Functions can again be then def classified as injections (or 1:1) and surjections (or A f −f(A) = sat(A) onto). f: X Y is said to be injective (or one-to- ⊂ one) if x = →x f(x ) = f(x ) for all x , x X, where sat(A) is the saturation of A X induced by 1 2 1 2 1 2 ⊆ while it is6 surje⇒ctive (or6 onto) if Y = f(X).∈f is f; if f is not surjective then bijective if it is both 1:1 and onto. ff −(B) := B f(X) B. Associated with a function f: X Y is its in- ⊆ → verse f −1: Y (f) X that exists on (f) iff If A = sat(A), then A is saidT to be saturated, and ⊇ R → R f is injective. Thus when f is bijective, f −1(y) := B (f) whenever ff −(B) = B. Thus for non- x X: y = f(x) exists for every y Y ; infact f is injectiv⊆ Re f, f −f is not an identity on X just as { ∈ } ∈ bijective iff f −1( y ) is a singleton for each y Y . ff − is not 1 if f is not surjective. However the { } ∈ Y Non-injective functions are not at all rare; if any- set of relations thing, they are very common even for linear maps − − − − and it would be perhaps safe to conjecture that ff f = f, f ff = f (1) they are overwhelmingly predominant in the non- that is always true will be of basic significance in linear world of nature. Thus for example, the simple this work. Following are some equivalent statements

3We do not distinguish between a relation and its graph although technically they are diﬀerent objects. Thus although a functional relation, strictly speaking, is the triple (X, f, Y ) written traditionally as f: X Y , we use it synonymously with → the graph f itself. Parenthetically, the word functional in this paper is not necessarily employed for a scalar-valued function, but is used in a wider sense to distinguish between a function and an arbitrary relation (that is a multifunction). Formally, whereas an arbitrary relation from X to Y is a subset of X Y , a functional relation must satisfy an additional restriction × that requires y = y whenever (x, y ) f and (x, y ) f. In this subset notation, (x, y) f y = f(x). 1 2 1 ∈ 2 ∈ ∈ ⇔ December 3, 2003 12:13 00851

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on the injectivity and surjectivity of functions f: set of X under , denoted by X/ := [x]: x X , X Y . has the equivalence∼ classes [x] as ∼its elemen{ ts;∈thus} → [x] plays a dual role either as subsets of X or as ele- (Injec) f is 1:1 there is a function fL: Y X ⇔ → ments of X/ . The rule x [x] defines a surjective called the left inverse of f, such that fLf = 1X ∼ 7→ A = f −f(A) for all subsets A of X f( A ) ⇔= function Q: X X/ known as the quotient map. ⇔ i → ∼ f(Ai). T Example 1.1. Let (Surjec) f is onto there is a function fR: Y X T ⇔ → S1 = (x, y) R2) : x2 + y2 = 1 called the right inverse of f, such that ffR = 1Y { ∈ } B = ff −(B) for all subsets B of Y . ⇔ be the unit circle in R2. Consider X = [0, 1] as a As we are primarily concerned with non- subspace of R, define a map injectivity of functions, saturated sets generated by q : X S1, s (cos 2πs, sin 2πs), s X , equivalence classes of f will play a significant role → 7→ ∈ in our discussions. A relation on a set X is said from R to R2, and let be the equivalence relation E 4 to be an equivalence relation if it is on X ∼ (ER1) Reflexive: ( x X)(x x). ∀ ∈ E s t (s = t) (s = 0, t = 1) (s = 1, t = 0) . (ER2) Symmetric: ( x, y X)(x y y x). ∼ ⇔ ∨ ∨ ∀ ∈ E ⇒ E (ER3) Transitive: ( x, y, z X)(x y y z If we bend X around till its ends touch, the resulting ∀ ∈ E ∧ E ⇒ x z). circle represents the quotient set Y = X/ whose E points are equivalent under as follows ∼ Equivalence relations group together unequal ele- ∼ ments x1 = x2 of a set as equivalent according to [0] = 0, 1 = [1], [s] = s for all s (0, 1) . the requiremen6 ts of the relation. This is expressed { } { } ∈ as x x (mod ) and will be represented here by Thus q is bijective for s (0, 1) but two-to-one for 1 2 ∈ the shorthand∼ notationE x x , or even simply the special values s = 0 and 1, so that for s, t X, 1 E 2 ∈ as x x if the specification∼ of is not essential. 1 ∼ 2 E s t q(s) = q(t) . Thus for a non-injective map if f(x1) = f(x2) for ∼ ⇔ x1 = x2, then x1 and x2 can be considered to be This yields a bijection h: X/ S1 such that equiv6 alent to each other since they map onto the ∼ → same point under f; thus x x f(x ) = q = h Q 1 ∼f 2 ⇔ 1 ◦ f(x2) defines the equivalence relation f induced defines the quotient map Q: X X/ by h([s]) = by the map f. Given an equivalence relation∼ on → ∼ ∼ q(s) for all s [0, 1]. The situation is illustrated by a set X and an element x X the subset the commutativ∈ e diagram of Fig. 2 that appears as def ∈ [x] = y X : y x an integral component in a different and more gen- { ∈ ∼ } is called the equivalence class of x; thus x y eral context in Sec. 2. It is to be noted that com- [x] = [y]. In particular, equivalence classes∼gener-⇔ mutativity of the diagram implies that if a given ated by f: X Y , [x] = x X : f(x ) = equivalence relation on X is completely deter- → f { α ∈ α mined by q that associates∼ the partitioning equiva- f(x) , will be a cornerstone of our analysis of chaos 1 generated} by the iterates of non-injective maps, and lence classes in X to unique points in S , then is identical to the equivalence relation that is induced∼ the equivalence relation f := (x, y): f(x) = f(y) generated by f is uniquely∼ defined{ by the partition} by Q on X. Note that a larger size of the equivalence R that f induces on X. Of course as x x, x [x]. classes can be obtained by considering X = + for ∼ ∈ which s t s t Z . It is a simple matter to see that any two equiva- ∼ ⇔ | − | ∈ + lence classes are either disjoint or equal so that the equivalence classes generated by an equivalence re- End Tutorial 1 lation on X form a disjoint cover of X. The quotient

4An alternate useful way of expressing these properties for a relation on X are R (ER1) is reflexive iff 1X X R ⊆ (ER2) is symmetric iff = −1 R R R (ER3) is transitive iff , R R ◦ R ⊆ R with an equivalence relation only if = . R R ◦ R R December 3, 2003 12:13 00851

3152 A. Sengupta

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¡¥¤§¦ ¨ © £ of , while (x) denotes the possible precursors M M + of x in X1 (or of X2) and (B) is that subset of X whose image lies in B forMany subset B X. ⊂ Fig. 2. The quotient map Q. In this paper the multifunctions that we shall be explicitly concerned with arise as the inverses of non-injective maps. One of the central concepts that we consider and The second major component of our theory is employ in this work is the inverse f − of a nonlin- the graphical convergence of a net of functions to ear, non-injective, function f; here the equivalence a multifunction. In Tutorial 2 below, we replace for classes [x] = f −f(x) of x X are the saturated f the sake of simplicity and without loss of generality, subsets of X that partition∈X. While a detailed the net (which is basically a sequence where the in- treatment of this question in the form of the non- dex set is not necessarily the positive integers; thus linear ill-posed problem and its solution is given in every sequence is a net but the family5 indexed, for Sec. 2 [Sengupta, 1997], it is suﬃcient to point out example, by Z, the set of all integers, is a net and here from Figs. 1(c) and 1(d), that the inverse of a not a sequence) with a sequence and provide the non-injective function is not a function but a mul- necessary background and motivation for the con- tifunction while the inverse of a multifunction is a cept of graphical convergence. non-injective function. Hence one has the general result that f is a non-injective function − f is a multifunction . Begin Tutorial 2: Convergence of ⇔ (2) f is a multifunction Functions f − is a non-injective function ⇔ This Tutorial reviews the inadequacy of the usual notions of convergence of functions either to limit The inverse of a multifunction : X– Y is a gen- M → functions or to distributions and suggests the mo- eralization of the corresponding notion for a func- tivation and need for introduction of the notion tion f: X Y such that → of graphical convergence of functions to multifunc- − def tions. Here, we follow closely the exposition of (y) = x X : y (x) ∞ M { ∈ ∈ M } Korevaar [1968], and use the notation (fk)k=1 to de- leads to note real or complex valued functions on a bounded or unbounded interval J. −(B) = x X : (x) B = M { ∈ M 6 ∅} A sequence of piecewise continuous functions ∞ for any B Y , while a more restricted\ inverse (fk)k=1 is said to converge to the function f, nota- ⊆ that we shall not be concerned with is given as tion fk f, on a bounded or unbounded interval 6 → +(B) = x X : (x) B . Obviously, J M+ { − ∈ M ⊆ } (B) (B). A multifunction is injective if (1) Pointwise if Mx = x ⊆ M (x ) (x ) = , and commonly 1 6 2 ⇒ M 1 M 2 ∅ with functions, it is true that Aα = fk(x) f(x) for all x J , T M α∈D → ∈ 5A function χ: D X will be called a familyS in X indexed by D when reference to the domain D is of interest, and a net → when it is required to focus attention on its values in X. 6 Observe that it is not being claimed that f belongs to the same class as (fk). This is the single most important cornerstone on which this paper is based: the need to “complete” spaces that are topologically “incomplete”. The classical high-school example of the related problem of having to enlarge, or extend, spaces that are not big enough is the solution space of algebraic equations with real coeﬃcients like x2 + 1 = 0. December 3, 2003 12:13 00851

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i.e. Given any arbitrary real number ε > 0 there It is to be observed that apart from point- exists a K N that may depend on x, such that wise and uniform convergences, all the other modes f (x) f(x∈) < ε for all k K. listed above represent some sort of an averaged con- | k − | ≥ (2) Uniformly if tribution of the entire interval J and are therefore not of much use when pointwise behavior of the sup f(x) fk(x) 0 as k , limit f is necessary. Thus while limits in the mean x∈J | − | → → ∞ are not unique, oscillating functions are tamed by i.e. Given any arbitrary real number ε > 0 there m-integral convergence for adequately large values exists a K N, such that sup f (x) f(x) < ε ∈ x∈J | k − | of m, and convergence relative to test functions, for all k K. as we see below, can be essentially reduced to m- ≥ p (3) In the mean of order p 1 if f(x) fk(x) is integral convergence. On the contrary, our graphical integrable over J for each k≥ | − | convergence — which may be considered as a pointwise biconvergence with respect to both the direct f(x) f (x) p 0 as k . and inverse images of f just as usual pointwise con- | − k | → → ∞ ZJ vergence is with respect to its direct image only For p = 1, this is the simple case of convergence in — allows a sequence (in fact, a net) of functions to the mean. converge to an arbitrary relation, unhindered by ex- (4) In the mean m-integrally if it is possible to select ternal influences such as the effects of integrations indefinite integrals and test functions. To see how this can indeed matter, consider the following x x1 (−m) f (x) = πk(x) + dx1 dx2 k Example 1.2. Let fk(x) = sin kx, k = 1, 2, . . . and Zc Zc xm−1 let J be any bounded interval of the real line. Then dxmfk(xm) 1-integrally we have · · · c x Z (−1) 1 1 and fk (x) = cos kx = + sin kx1dx1 , −k −k 0 x x1 Z (−m) f (x) = π(x) + dx1 dx2 which obviously converges to 0 uniformly (and Zc Zc therefore in the mean) as k . And herein lies − → ∞ xm 1 the point: even though we cannot conclude about dxmf(xm) · · · c the exact nature of sin kx as k increases indefi- Z nitely (except that its oscillations become more and such that for some arbitrary real p 1, ≥ more pronounced), we may very definitely state that (−m) lim (cos kx)/k = 0 uniformly. Hence from f (−m) f p 0 as k . k→∞ | − k | → → ∞ x ZJ (−1) f (x) 0 = 0 + lim sin kx1dx1 where the polynomials π (x) and π(x) are of degree k → k→∞ k Z0 < m, and c is a constant to be chosen appropriately. it follows that (5) Relative to test functions ϕ if fϕ and f ϕ are k lim sin kx = 0 (3) integrable over J and k→∞ 1-integrally. ∞ (fk f)ϕ 0 for every ϕ 0 (J) as k , Continuing with the same sequence of func- J − → ∈ C → ∞ Z tions, we now examine its test-functional conver- where ∞(J) is the class of infinitely differentiable 1 C0 gence with respect to ϕ 0 ( , ) that vanishes continuous functions that vanish throughout some for all x / (α, β). Integrating∈ C −∞by parts,∞ neighborhood of each of the end points of J. For ∈ ∞ β an unbounded J, a function is said to vanish in fkϕ = ϕ(x1) sin kx1dx1 some neighborhood of + if it vanishes on some −∞ α ∞ Z Z ray (r, ). 1 ∞ = [ϕ(x ) cos kx ]β While pointwise convergence does not imply −k 1 1 α any other type of convergence, uniform conver- 1 β gence on a bounded interval implies all the other ϕ0(x ) cos kx dx −k 1 1 1 convergences. Zα December 3, 2003 12:13 00851

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Fig. 3. Incompleteness of function spaces. (a) demonstrates the classic example of non-completeness of the space of real- valued continuous functions leading to the complete spaces Ln[a, b] whose elements are equivalence classes of functions with b f g iﬀ the Lebesgue integral f g n = 0. (b) and (c) illustrate distributional convergence of the functions f (x) of ∼ a | − | k Eq. (5) to the Dirac delta δ(x) leading to the complete space of generalized functions. In comparison, note that the space R of continuous functions in the uniform metric C[a, b] is complete which suggests the importance of topologies in determining convergence properties of spaces.

The first integrated term is 0 due to the condi- converges in the mean to f (−m)ϕ(m) so that tions on ϕ while the second also vanishes because β β 1 m (−m) (m) ϕ 0 ( , ). Hence f ϕ = ( 1) f ϕ ∈ C −∞ ∞ k k α − α ∞ β Z Z β β fkϕ 0 = lim ϕ(x1) sin ksdx1 → k→∞ ( 1)m f (−m)ϕ(m) = fϕ. Z−∞ Zα → − Zα Zα for all ϕ, and leading to the conclusion that In fact the converse also holds leading to the following Equivalences between m-convergence in lim sin kx = 0 (4) k→∞ the mean and convergence with respect to test- functions [Korevaar, 1968]. test-functionally. Type 1 Equivalence. If f and (f ) are functions This example illustrates the fact that if k on J that are integrable on every interior subinter- Supp(ϕ) = [α, β] J,7 integrating by parts suf- val, then the following are equivalent statements. ficiently large num⊆ber of times so as to wipe out the pathological behavior of (fk) gives (a) For every interior subinterval I of J there is an integer mI 0, and hence a smallest in- β teger m 0, suc≥h that certain indefinite inte- fkϕ = fkϕ (−≥m) ZJ Zα grals fk of the functions fk converge in the (−m) β β mean on I to an indefinite integral f ; thus (−1) (−m) 0 m m (−m) (−m) = fk ϕ = = ( 1) fk ϕ f f 0. α · · · − α I k Z Z | − | → ∞ (b) J (fk f)ϕ 0 for every ϕ 0 (J). (−m) x x1 xm−1 R − → ∈ C where f (x) = π (x) + dx dx k k c 1 c 2 c A significanR t generalization of this Equivalence is dx f (x ) is an m-times arbitrary indefinite· · · in- m k m R R R obtained by dropping the restriction that the limit tegral of f . If now it is true that β f (−m) k α k → object f be a function. The need for this gener- β (−m) (−m) (m) α f , then it must also be true thatR fk ϕ alization arises because metric function spaces are R 7By definition, the support (or supporting interval) of ϕ(x) ∞[α, β] is [α, β] if ϕ and all its derivatives vanish for x α ∈ C0 ≤ and x β. ≥ December 3, 2003 12:13 00851

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known not to be complete: Consider the sequence can be associated with the arbitrary indefinite of functions [Fig. 3(a)] integrals 0, if a x 0 0, a x 0 ≤ ≤ ≤ ≤  1 1 fk(x) = kx, if 0 x (5) def (−1)  kx, 0 < x <  ≤ ≤ k Θk(x) = δk (x) =    k  1  1 1, if x b 1, x b k ≤ ≤ k ≤ ≤   which is not Cauc hy in the uniform metric   of Fig. 3(c), which, as noted above, converge ρ(fj, fk) = supa≤x≤b fj(x) fk(x) but is Cauchy | − | in the mean to the unit step function Θ(x); in the mean ρ(f , f ) = b f (x) f (x) dx, or j k a j k ∞ β β (−1) 0 | − | hence δkϕ δkϕ = δ ϕ even pointwise. However in either case, (fk) cannot −∞ ≡ α − α k → converge in the respective Rmetrics to a continuous β ϕ0(x)dx = ϕ(0). But there can be no func- − 0 R R R function and the limit is a discontinuous unit step β tionalR relation δ(x) for which α δ(x)ϕ(x)dx = ϕ(0) function 1 for all ϕ C0 [α, β], so that unlike in the case in 0, if a x 0 Type 1 Equiv∈ alence, the limitR in the mean Θ(x) Θ(x) = ≤ ≤ (−1) ( 1, if 0 < x b of the indefinite integrals δk (x) cannot be ex- ≤ pressed as the indefinite integral δ(−1)(x) of some with graph ([a, 0], 0) ((0, b], 1), which is also in- function δ(x) on any interval containing the ori- tegrable on [a, b]. Thus even if the limit of the se- gin. This leads to the second more general type of quence of continuousSfunctions is not continuous, equivalence. both the limit and the members of the sequence are integrable functions. This Riemann integration Type 2 Equivalence. If (f ) are functions on J is not sufficiently general, however, and this type k that are integrable on every interior subinterval, of integrability needs to be replaced by a much then the following are equivalent statements. weaker condition resulting in the larger class of the Lebesgue integrable complete space of functions (a) For every interior subinterval I of J there is an 8 L[a, b]. integer mI 0, and hence a smallest integer The functions in Fig. 3(b), m 0, suc≥h that certain indefinite integrals ≥ 1 f (−m) of the functions f converge in the mean k, if 0 < x < k k k on I to an integrable function Θ which, unlike δ (x) =  k 1 in Type 1 Equivalence, need not itself be an  0, x [a, b] 0, ,  ∈ − k indefinite integral of some function f.  8Both Riemann and Lebesgue integrals can be formulated in terms of the so-called step functions s(x), which are piecewise I I constant functions with values (σi)i=1on a finite number of bounded subintervals (Ji)i=1 (which may reduce to a point or def I may not contain one or both of the end points) of a bounded or unbounded interval J, with integral s(x)dx = σi Ji . J i=1 | | While the Riemann integral of a bounded function f(x) on a bounded interval J is defined with respect to sequences R P ∞ ∞ of step functions (sj) and (tj ) satisfying sj (x) f(x) tj (x) on J with (sj tj ) 0 as j as j=1 j=1 ≤ ≤ J − → → ∞ R J f(x)dx = lim J sj (x)dx = lim J tj(x)dx, the less restrictive Lebesgue integral isR defined for arbitrary functions f over bounded or unbounded intervals J in terms of Cauchy sequences of step functions si s 0, i, k , converging R R R J | − k| → → ∞ to f(x) as R

sj (x) f(x) pointwise almost everywhere on J , → to be def f(x)dx = lim sj(x)dx . j→∞ ZJ ZJ That the Lebesgue integral is more general (and therefore is the proper candidate for completion of function spaces) is illustrated by the example of the function defined over [0, 1] to be 0 on the rationals and 1 on the irrationals for which an application of the definitions verify that while the Riemann integral is undefined, the Lebesgue integral exists and has value 1. The Riemann integral of a bounded function over a bounded interval exists and is equal to its Lebesgue integral. Because it involves a larger family of functions, all integrals in integral convergences are to be understood in the Lebesgue sense. December 3, 2003 12:13 00851

3156 A. Sengupta

∞ system evolves to a state of maximal ill-posedness. (b) ck(ϕ) = J fkϕ c(ϕ) for every ϕ 0 (J). → ∈ C The analysis is based on the non-injectivity, and (−m) Since we areR now given that f (x)dx hence ill-posedness, of the map; this may be viewed I k → (−m) (m) as a mathematical formulation of the stretch-and- I Ψ(x)dx, it must also be true thatR fk ϕ converges in the mean to Ψϕ(m) whence fold and stretch-cut-and-paste kneading operations R of the dough that are well-established artifacts in f ϕ = ( 1)m f (−m)ϕ(m) the theory of chaos and the concept of maximal ill- k − k ZJ ZI posedness helps in obtaining a physical understanding of the nature of chaos. We do this through the m (m) m (−m) (m) ( 1) Ψϕ = ( 1) f ϕ . fundamental concept of the graphical convergence of → − I 6 − I Z Z a sequence (generally a net) of functions [Sengupta The natural question that arises at this stage is & Ray, 2000] that is allowed to converge graphically, then: What is the nature of the relation (not func- when the conditions are right, to a set-valued map tion any more) Ψ(x)? For this it is now stipulated, or multifunction. Since ill-posed problems naturally despite the non-equality in the equation above, that lead to multifunctional inverses through functional as in the mean m-integral convergence of (fk) to a generalized inverses [Sengupta, 1997], it is natural function f, to seek solutions of ill-posed problems in multifunc- x (−1) def 0 0 tional space Multi(X, Y ) rather than in spaces of Θ(x) := lim δk (x) = δ(x )dx (6) k→∞ −∞ functions Map(X, Y ); here Multi(X, Y ) is an ex- Z tension of Map(X, Y ) that is generally larger than defines the non-functional relation (“generalized the smallest dense extension Multi (X, Y ). function”) δ(x) integrally as a solution of the inte- | 9 Feedback and iteration are natural processes by gral equation (6) of the first kind; hence formally which nature evolves itself. Thus almost every pro- dΘ cess of evolution is a self-correction process by which δ(x) = (7) dx the system proceeds from the present to the future through a controlled mechanism of input and eval- End Tutorial 2 uation of the past. Evolution laws are inherently nonlinear and complex; here complexity is to be understood as the natural manifestation of the non- The above tells us that the “delta function” is not linear laws that govern the evolution of the system. a function but its indefinite integral is the piecewise This paper presents a mathematical description continuous function Θ obtained as the mean (or of complexity based on [Sengupta, 1997] and [Sen- pointwise) limit of a sequence of non-differentiable gupta & Ray, 2000] and is organized as follows. functions with the integral of dΘk(x)/dx being pre- In Sec. 1, we follow [Sengupta, 1997] to give an served for all k Z+. What then is the delta overview of ill-posed problems and their solution (and not its integral)?∈ The answer to this ques- that forms the foundation of our approach. Sec- tion is contained in our multifunctional extension tions 2 to 4 apply these ideas by defining a chaotic Multi(X, Y ) of the function space Map(X, Y ) con- dynamical system as a maximally ill-posed problem; sidered in Sec. 3. Our treatment of ill-posed prob- by doing this we are able to overcome the limi- lems is used to obtain an understanding and inter- tations of the three Devaney characterizations of pretation of the numerical results of the discretized chaos [Devaney, 1989] that apply to the specific case spectral approximation in neutron transport the- of iteration of transformations in a metric space, ory [Sengupta, 1988, 1995]. The main conclusions and the resulting graphical convergence of func- are the following: In a one-dimensional discrete sys- tions to multifunctions is the basic tool of our ap- tem that is governed by the iterates of a nonlin- proach. Section 5 analyzes graphical convergence in ear map, the dynamics is chaotic if and only if the Multi(X) for the discretized spectral approximation

9The observant reader cannot have failed to notice how mathematical ingenuity successfully transferred the “troubles” of ∞ (δk)k=1 to the sufficiently differentiable benevolent receptor ϕ so as to be able to work backward, via the resultant trouble free (−m) ∞ (δk )k=1, to the final object δ. This necessarily hides the true character of δ to allow only a view of its integral manifestation on functions. This unfortunately is not general enough in the strongly nonlinear physical situations responsible for chaos, and is the main reason for constructing the multifunctional extension of function spaces that we use. December 3, 2003 12:13 00851

Toward a Theory of Chaos 3157

of neutron transport theory, which suggests a nat- Example 2.1. As a non-trivial example of an in- ural link between ill-posed problems and spectral verse problem, consider the heat equation theory of nonlinear operators. This seems to offer ∂θ(x, t) ∂2θ(x, t) an answer to the question of why a natural sys- = c2 tem should increase its complexity, and eventually ∂t ∂x2 tend toward chaoticity, by becoming increasingly for the temperature distribution θ(x, t) of a one- nonlinear. dimensional homogeneous rod of length L satisfying the initial condition θ(x, 0) = θ0(x), 0 x L, 2. Ill-Posed Problem and Its and boundary conditions θ(0, t) = 0 = θ(L,≤ t),≤0 Solution t T , having the Fourier sine-series solution ≤ ≤ This section based on [Sengupta, 1997] presents ∞ nπ 2 a formulation and solution of ill-posed problems θ(x, t) = A sin x e−λnt (8) n L arising out of the non-injectivity of a function f: n=1 X X Y between topological spaces X and Y . A where λ = (cπ/a)n and work→able knowledge of this approach is necessary as n a our theory of chaos leading to the characterization 2 0 nπ 0 0 An = θ0(x ) sin x dx of chaotic systems as being a maximally ill-posed L 0 L state of a dynamical system is a direct application of Z these ideas and can be taken to constitute a math- are the Fourier expansion coefficients. While the di- ematical representation of the familiar stretch-cut- rect problem evaluates θ(x, t) from the differential and paste and stretch-and-fold paradigms of chaos. equation and initial temperature distribution θ0(x), The problem of finding an x X for a given y Y the inverse problem calculates θ0(x) from the inte- from the functional relation f∈(x) = y is an inv∈erse gral equation problem that is ill-posed (or, the equation f(x) = y 2 a θ (x) = k(x, x0)θ (x0)dx0, 0 x L, is ill-posed) if any one or more of the following con- T L 0 ≤ ≤ ditions are satisfied. Z0 when this final temperature θ is known, and (IP1) f is not injective. This non-uniqueness prob- T ∞ lem of the solution for a given y is the single most nπ nπ 2 k(x, x0) = sin x sin x0 e−λnT significant criterion of ill-posedness used in this L L n=1 work. X (IP2) f is not surjective. For a y Y , this is the ∈ is the kernel of the integral equation. In terms of existence problem of the given equation. the final temperature the distribution becomes (IP3) When f is bijective, the inverse f −1 is not ∞ continuous, which means that small changes in y nπ 2 θ (x) = B sin x e−λn(t−T ) (9) may lead to large changes in x. T n L n=1 X A problem f(x) = y for which a solution exists, with Fourier coefficients is unique, and small changes in data y that lead a 2 0 nπ 0 0 to only small changes in the solution x is said to Bn = θT (x ) sin x dx . L 0 L be well-posed or properly posed. This means that Z f(x) = y is well-posed if f is bijective and the In L2[0, a], Eqs. (8) and (9) at t = T and t = 0 inverse f −1: Y X is continuous; otherwise the yield respectively → equation is ill-posed or improperly posed. It is to ∞ be noted that the three criteria are not, in general, 2 L 2 −2λ2 T −2λ2T 2 θ (x) = A e n e 1 θ (10) independent of each other. Thus if f represents a k T k 2 n ≤ k 0k n=1 bijective, bounded linear operator between Banach X ∞ spaces X and Y , then the inverse mapping theo- L 2 θ 2 = B2e2λnT . (11) rem guarantees that the inverse f −1 is continuous. k 0k 2 n n=1 Hence ill-posedness depends not only on the alge- X braic structures of X, Y , f but also on the topolo- The last two equations differ from each other in gies of X and Y . the significant respect that whereas Eq. (10) shows December 3, 2003 12:13 00851

3158 A. Sengupta

that the direct problem is well-posed according to (b) For a linear operator A: Rn Rm, m < n, sat- (IP3), Eq. (11) means that in the absence of similar isfying (1) and (2), the problem→Ax = y reduces A bounds the inverse problem is ill-posed.10 to echelon form with rank r less than min m, n , when the given equations are consistent. The{ solu-} tion however, produces a generalized inverse leading Example 2.2. Consider the Volterra integral equa- to a set-valued inverse A− of A for which the inverse tion of the first kind images of y (A) are multivalued because of the ∈ R x non-trivial null space of A introduced by assump- y(x) = r(x0)dx0 = Kr tion (1). Specifically, a null-space of dimension n r n − Za is generated by the free variables xj j=r+1 which are arbitrary: this is illposedness of{typ}e (1). In ad- where y, r C[a, b] and K: C[0, 1] C[0, 1] is ∈ → dition, m r rows of the row reduced echelon form the corresponding integral operator. Since the dif- of A have−all 0 entries that introduce restrictions ferential operator D = d/dx under the sup-norm m on m r coordinates yi i=r+1 of y which are now r = sup r(x) is unbounded, the inverse − r { } k k 0≤x≤1 | | related to yi i=1: this illustrates ill-posedness of problem r = Dy for a differentiable function y type (2). In{verse} ill-posed problems therefore gen- on [a, b] is ill-posed, see Example 6.1. However, erate multivalued solutions through a generalized y = Kr becomes well-posed if y is considered to be inverse of the mapping. in C1[0, 1] with norm y = sup Dy . This il- 0≤x≤1 (c) The eigenvalue problem lustrates the importancek kof the topologies| |of X and Y in determining the ill-posed nature of the prob- d2 lem when this is due to (IP3). + λ2 y = 0 y(0) = 0 = y(1) dx2 Ill-posed problems in nonlinear mathematics of type (IP1) arising from the non-injectivity of f has the following equivalence class of 0 can be considered to be a generalization of non- 2 ∞ 2 d 2 uniqueness of solutions of linear equations as, for [0] 2 = sin(πmx) , D = + λ , D { }m=0 dx2 example, in eigenvalue problems or in the solution of a system of linear algebraic equations with a larger as its eigenfunctions corresponding to the eigenval- number of unknowns than the number of equations. ues λ = πm. In both cases, for a given y Y , the solution set of m Ill-posed problems are primarily of interest to the equation f(x) = y is giv∈en by us explicitly as non-injective maps f, that is under − 0 0 f (y) = [x]f = x X : f(x ) = f(x) = y . the condition of (IP1). The two other conditions { ∈ } (IP2) and (IP3) are not as significant and play only A significant point of difference between linear and an implicit role in the theory. In its application to nonlinear problems is that unlike the special im- iterative systems, the degree of non-injectivity of f portance of 0 in linear mathematics, there are no defined as the number of its injective branches, in- preferred elements in nonlinear problems; this leads creases with iteration of the map. A necessary (but to a shift of emphasis from the null space of linear not sufficient) condition for chaos to occur is the problems to equivalence classes for nonlinear equa- increasing non-injectivity of f that is expressed de- tions. To motivate the role of equivalence classes, scriptively in the chaos literature as stretch-and-fold let us consider the null spaces in the following lin- or stretch-cut-and-paste operations. This increasing ear problems. non-injectivity that we discuss in the following sec- (a) Let f: R2 R be defined by f(x, y) = x + y, tions, is what causes a dynamical system to tend (x, y) R2. The→ null space of f is generated by the toward chaoticity. Ill-posedness arising from non- equation∈ y = x on the x–y plane, and the graph surjectivity of (injective) f in the form of regular- of f is the plane− passing through the lines ρ = x ization [Tikhonov & Arsenin, 1977] has received and ρ = y. For each ρ R the equivalence classes wide attention in the literature of ill-posed prob- f −(ρ) = (x, y) R2: ∈x + y = ρ are lines on the lems; this however is not of much significance in graph parallel{ to∈the null set. } our work.

10Recall that for a linear operator continuity and boundedness are equivalent concepts. December 3, 2003 12:13 00851

Toward a Theory of Chaos 3159

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Fig. 4. (a) Moore–Penrose generalized inverse. The decomposition of X and Y into the four fundamental subspaces of A comprising the null space (A), the column (or range) space (A), the row space (AT) and (AT), the complement of N R R N (A) in Y , is a basic result in the theory of linear equations. The Moore–Penrose inverse takes advantage of the geometric R orthogonality of the row space (AT) and (A) in Rn and that of the column space and (AT) in Rm. (b) When X and R N N Y are not inner-product spaces, a non-injective inverse can be deﬁned by extending f to Y (f) suitably as shown by − R the dashed curve, where g(x) := r + ((r r )/r )f(x) for all x (f) was taken to be a good deﬁnition of an extension 1 2 − 1 1 ∈ D that replicates f in Y (f); here x x under both f and g, and y y under f, g just as b is equivalent to − R 1 ∼ 2 1 ∼ 2 { } b in the Moore–Penrose case. Note that both f, g and f −, g− are both multifunctions on X and Y , respectively. Our k { } { } inverse G, introduced later in this section, is however injective with G(Y (f)) := 0. − R

map a) is the noninjective map defined in terms of the row and column spaces of A, row(A) = (AT), Begin Tutorial 3: Generalized col(A) = (A), as R Inverse R −1 In this Tutorial, we take a quick look at the equation def (a row(A)) (y), if y col(A) a(x) = y, where a: X Y is a linear map that need GMP(y) = | ∈ T ( 0, if y (A ) . not be either one-one→or onto. Specifically, we will ∈ N take X and Y to be the Euclidean spaces Rn and (12) Rm so that a has a matrix representation A Rm×n m×n ∈ Note that the restriction a of a to (AT) where R is the collection of m n matrices with |row(A) R −1 × −1 real entries. The inverse A exists and is unique iff is bijective so that the inverse (a row(A)) is well- | m = n and rank(A) = n; this is the situation de- defined. The role of the transpose matrix appears picted in Fig. 1(a). If A is neither one-one or onto, naturally, and the GMP of Eq. (12) is the unique then we need to consider the multifunction A−, a matrix that satisfies the conditions functional choice of which is known as the generalized inverse G of A. A good introductory text for AGMPA = A, GMPAGMP = GMP, T T (13) generalized inverses is [Campbell & Mayer, 1979]. (GMPA) = GMPA, (AGMP) = AGMP Figure 4(a) introduces the following definition of the Moore–Penrose generalized inverse GMP. that follow immediately from the definition (12); hence GMPA and AGMP are orthogonal projec- 11 T Definition 2.1 (Moore–Penrose Inverse). If a: tions onto the subspaces (A ) = (GMP) and Rn Rm is a linear transformation with matrix (A), respectively. Recall Rthat the Rrange space represen→ tation A Rm×n then the Moore–Penrose R(AT) of AT is the same as the row space row(A) ∈n×m R inverse GMP R of A (we will use the same of A, and (A) is also known as the column space notation G ∈: Rm Rn for the inverse of the of A, col(AR). MP →

11A real matrix A is an orthogonal projector iﬀ A2 = A and A = AT. December 3, 2003 12:13 00851

3160 A. Sengupta

Example 2.3. For a: R5 R4, let rank is 4. This gives → 9 1 18 2 1 3 2 1 2 − 275 −275 275 −55 3 9 10 2 9  27 3 54 6  A =  −   −275 275 −275 55  2 6 4 2 4    −   10 6 20 16   2 6 8 1 7  G =    −  MP      −143 143 −143 143   238 57 476 59    By reducing the augmented matrix (A y) to the   |  3575 −3575 3575 −715  row-reduced echelon form, it can be verified that    129 106 258 47  the null and range spaces of A are three- and two-   dimensional, respectively. A basis for the null space  −3575 3575 −3575 715   (14) of AT and of the row and column space of A obtained from the echelon form are respectively as the Moore–Penrose inverse of A that readily ver- ifies all the four conditions of Eqs. (13). The basic point here is that, as in the case of a bijective map, 1 0 GMPA and AGMP are identities on the row and col- 2 1  3   0  1 0 − − umn spaces of A that define its rank. For later use — 0 1 0 1 0 1             when we return to this example for a simpler inverse , − ; and  3  ,  1  ; , . 1 0     2 0 G — given below are the orthonormal bases of the      2   − 4       0   1       1   1           −    four fundamental subspaces with respect to which      1   3          G is a representation of the generalized inverse of  2   4  MP     A; these calculations were done by MATLAB. The basis for According to its definition Eq. (12), the Moore– Penrose inverse maps the middle two of the above (a) the column space of A consists of the first two set to (0, 0, 0, 0, 0)T, and the A-image of the first columns of the eigenvectors of AAT: two (which are respectively (19, 70, 38, 51)T and T 1633 363 3317 363 T (70, 275, 140, 205) lying, as they must, in the span , , , of the last two), to the span of (1, 3, 2, 1, 2)T and −2585 −892 6387 892 (3, 9, 10, 2, 9)T because a restricted− to this sub- − 929 709 346 709 T space of R5 is bijective. Hence , , , −1435 1319 6299 −1319 1 0 (b) the null space of AT consists of the last two 3 0 columns of the eigenvectors of AAT:   −    2 1  0 1 − T      0 1  3185 293 3185 1777 GMP A  3  A  1  −  , , ,      1 0  −8306 2493 −4153 3547   2   −4        0 1  T       323 533 323 1037   1   3   , , ,       1732 731 866 1911   2   4         1 0 0 0 (c) the row space of A consists of the first two 3 0 0 0 columns of the eigenvectors of ATA:  −  0 1 0 0 421 44 569 659 1036   =  3 1  . , , , ,  0 0  13823 14895 −918 −2526 1401  2 −4    661 412 59 1523 303    1 3  , , , ,  0 0  690 1775 2960 −10221 −3974  2 4    (d) the null space of A consists of the last three The second matrix on the left is invertible as its columns of ATA: December 3, 2003 12:13 00851

Toward a Theory of Chaos 3161

571 369 149 291 389 (T3) Arbitrary unions of members of belong , , , , U −15469 −776 25344 −350 −1365 to . U 281 956 875 1279 409 , , , , Example 2.4 −1313 1489 1706 −2847 1473 (1) The smallest topology possible on a set X is 292 876 203 621 1157 , , , , its indiscrete topology when the only open sets 1579 −1579 342 4814 2152 are and X; the largest is the discrete topology where∅ every subset of X is open (and hence also The matrices Q1 and Q2 with these eigenvectors closed). (xi) satisfying xi = 1 and (xi, xj) = 0 for i = j k k 6 (2) In a metric space (X, d), let B (x, d) = y X: as their columns are orthogonal matrices with the ε { ∈ simple inverse criterion Q−1 = QT. d(x, y) < ε be an open ball at x. Any subset U of X suc}h that for each x U there is a d- ∈ ball Bε(x, d) U in U, is said to be an open End Tutorial 3 set of (X, d).⊆The collection of all these sets is the topology induced by d. The topological space (X, ) is then said to be associated with U The basic issue in the solution of the inverse ill- (induced by) (X, d). posed problem is its reduction to an well-posed one (3) If is an equivalence relation on a set X, the set∼of all saturated sets [x] = y X: y x when restricted to suitable subspaces of the do- ∼ { ∈ ∼ } main and range of A. Considerations of geometry is a topology on X; this topology is called the leading to their decomposition into orthogonal sub- topology of saturated sets. spaces is only an additional feature that is not cen- We argue in Sec. 4.2 that this constitutes tral to the problem: recall from Eq. (1) that any the defining topology of a chaotic system. function f must necessarily satisfy the more general (4) For any subset A of the set X, the A-inclusion − − − − topology on X consists of and every superset set-theoretic relations ff f = f and f ff = f ∅ of Eq. (13) for the multiinverse f − of f: X Y . of A, while the A-exclusion topology on X con- → sists of all subsets of X A. Thus A is open The second distinguishing feature of the MP-inverse − is that it is defined, by a suitable extension, on all in the inclusion topology and closed in the ex- of Y and not just on f(X) which is perhaps more clusion, and in general every open set of one is natural. The availability of orthogonality in inner- closed in the other. product spaces allows this extension to be made The special cases of the a-inclusion and a- exclusion topologies for A = a are defined in in an almost normal fashion. As we shall see be- { } low the additional geometric restriction of Eq. (13) a similar fashion. is not essential to the solution process, and in- (5) The cofinite and cocountable topologies in which fact, only results in a less canonical form of the the open sets of an infinite (resp. uncount- inverse. able) set X are respectively the complements of finite and countable subsets, are examples of topologies with some unusual properties that are covered in Appendix A.1. If X is itself finite (respectively, countable), then its cofinite Begin Tutorial 4: Topological Spaces (respectively, cocountable) topology is the dis- This Tutorial is meant to familiarize the reader with crete topology consisting of all its subsets. It is the basic principles of a topological space. A topo- therefore useful to adopt the convention, unless logical space (X, ) is a set X with a class12 of U U stated to the contrary, that cofinite and co- distinguished subsets, called open sets of X, that countable spaces are respectively infinite and satisfy uncountable. (T1) The empty set and the whole X belong to ∅ U In the space (X, ), a neighborhood of a point (T2) Finite intersections of members of belong x X is a nonemptUy subset N of X that con- to U tains∈ an open set U containing x; thus N X is a U ⊆

12In this sense, a class is a set of sets. December 3, 2003 12:13 00851

3162 A. Sengupta

neighborhood of x iff neighborhood system at x coincides exactly with x U N (15) the assigned collection x; compare with Defini- ∈ ⊆ tion A.1.1. NeighborhoodsN in topological spaces are for some U . The largest open set that can be ∈ U a generalization of the familiar notion of distances used here is Int(N) (where, by definition, Int(A) is of metric spaces that quantifies “closeness” of points the largest open set that is contained in A) so that of X. the above neighborhood criterion for a subset N of A neighborhood of a non-empty subset A of X X can be expressed in the equivalent form that will be needed later on is defined in a similar N X is a neighborhood of x iff x IntU (N) manner: N is a neighborhood of A iff A Int(N), ⊆ U − ∈ that is A U N; thus the neighborho⊆ od sys- (16) ⊆ ⊆ tem at A is given by A = a := G X: implying that a subset of (X, ) is a neighborhood N a∈A N { ⊆ U G a for every a A is the class of common of all its interior points, so that N x N y neigh∈ bNorhoods of each∈ poin} t Tof A. for all y Int(N). The collection∈ofNall⇒neigh∈bNor- ∈ Some examples of neighborhood systems at a hoods of x point x in X are the following: def = N X : x U N for some U Nx { ⊆ ∈ ⊆ ∈ U} (1) In an indiscrete space (X, ), X is the only (17) neighborhood of every pointUof the space; in a is the neighborhood system at x, and the subcol- discrete space any set containing x is a neigh- lection U of the topology used in this equation borhood of the point. constitutes a neighborhood (local ) base or basic (2) In an infinite cofinite (or uncountable cocount- neighborhood system, at x, see Definition A.1.1 of able) space, every neighborhood of a point is an Appendix A.1. The properties open neighborhood of that point. (3) In the topology of saturated sets under the (N1) x belongs to every member N of , Nx equivalence relation , the neighborhood sys- (N2) The intersection of any two neighborhoods of tem at x consists of all∼ supersets of the equiva- x is another neighborhood of x: N, M lence class [x] . ∈ Nx ⇒ ∼ N M , (4) Let x X. In the x-inclusion topology, ∈ Nx x consists∈of all the non-empty open sets ofNX (N3)T Every superset of any neighborhood of x is a neighborhood of x: (M ) (M N) N which are the supersets of x . For a point ∈ Nx ∧ ⊆ ⇒ ∈ { } , y = x of X, y are the supersets of x, y . Nx 6 N { } that characterize completely are a direct conse- For any given class T of subsets of X, a unique Nx S quence of the definitions (15), (16) that may also topology (T ) can always be constructed on X U S be stated as by taking all finite intersections T ∧ of members S of followed by arbitrary unions T ∧∨ of these fi- (N0) Any neighborhood N x contains another S S ∈ N nite intersections. (T ) := T ∧∨ is the smallest neighborhood U of x that is a neighborhood of each U S S topology on X that contains T and is said to be of its points: (( N x)( U x)(U N)) : S ∀ ∈ N ∃ ∈ N ⊆ generated by T . For a given topology on X satis- ( y U U y). S U ∀ ∈ ⇒ ∈ N fying = (T ), T is a subbasis, and T ∧ := T U U S S S B Property (N0) infact serves as the defining char- a basis, for the topology ; for more on topological U acteristic of an open set, and U can be identified basis, see Appendix A.1. The topology generated with the largest open set Int(N) contained in N; by a subbase essentially builds not from the collec- hence a set G in a topological space is open iff it is tion itself but from the finite intersections TS TS∧ a neighborhood of each of its points. Accordingly if of its subsets; in comparison the base generates a is a given class of subsets of X associated with topology directly from a collection of subsets x TS eacN h x X satisfying (N1)–(N3), then (N0) defines by forming their unions. Thus whereas any class of the special∈ class of neighborhoods G subsets can be used as a subbasis, a given collection = G : x B G for all x G must meet certain qualifications to pass the test of a U { ∈ Nx ∈ ⊆ ∈ base for a topology: these and related topics are cov- and some basic nbd B (18) ∈ Nx} ered in Appendix A.1. Different subbases, therefore, as the unique topology on X that contains a basic can be used to generate different topologies on the neighborhood of each of its points, for which the same set X as the following examples for the case of December 3, 2003 12:13 00851

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X = R demonstrates; here (a, b), [a, b), (a, b] and consisting of those points of X that are in A but [a, b], for a b R, are the usual open-closed inter- not in its boundary, Int(A) = A Bdy(A), is the 13≤ ∈ − vals in R. The subbases T 1 = (a, ), ( , b) , largest open subset of X that is contained in A. = [a, ), ( , b) , S ={ (a,∞ ), −∞( , b}] Hence it follows that Int(Bdy(A)) = , the bound- TS2 { ∞ −∞ } TS3 { ∞ −∞ } ∅ and T 4 = [a, ), ( , b] give the respective ary of A is the intersection of the closures of A and bases S = {(a, b∞) , −∞= }[a, b) , = (a, b] X A, and a subset N of X is a neighborhood of T 1 T 2 T 3 − and B= [{a, b] ,}a Bb R{, leading} Bto the{ stan-} x iff x Int(N). T 4 ∈ dard (Busual ){, lower} limit≤ (Sor∈ genfrey), upper limit, and discrete (take a = b) topologies on R. Bases of The three subsets Int(A), Bdy(A) and exterior the type (a, ) and ( , b) provide the right and of A defined as Ext(A) := Int(X A) = X Cl(A), − − left ray topologies∞ on R−∞. are pairwise disjoint and have the full space X as their union. This feasibility of generating different topologies on a set can be of great practi- Definition 2.3 (Derived and Isolated sets). Let A cal significance because open sets determine be a subset of X. A point x X (which may or convergence characteristics of nets and con- may not be a point of A) is a ∈cluster point of A if tinuity characteristics of functions, thereby every neighborhood N x contains at least one making it possible for nature to play around point of A different from∈ xN. The derived set of A with the structure of its working space in its 14 def kitchen to its best possible advantage. Der(A) = x X :( N x) N (A x )= ∈ ∀ ∈N −{ } 6 ∅ Here are a few essential concepts and terminology n \ (22)o for topological spaces. is the set of all cluster points of A. The complement of Der(A) in A Definition 2.2 (Boundary, Closure, Interior). The boundary of A in X is the set of points x X such Iso(A) def= A Der(A) = Cl(A) Der(A) (23) that every neighborhood N of x intersects∈ both A − − and X–A: are the isolated points of A to which no proper sequence in A converges, that is there exists a neigh- def Bdy(A) = x X : ( N x)((N A = borhood of any such point that contains no other { ∈ ∀ ∈ N 6 ∅ point of A so that the only sequence that converges (N (X A) = )) \ (19) ∧ − 6 ∅ } to a Iso(A) is the constant sequence (a, a, a, . . .). Clearly∈ , where is the neigh\ borhood system of Eq. (17) Nx at x. Cl(A) = A Der(A) = A Bdy(A) The closure of A is the set of all points x X such that each neighborhood of x contains at least∈ = Iso[(A) Der(A) [= Int(A) Bdy(A) one point of A that may be x itself. Thus the set with the last two[being disjoint unions,[ and A is def closed iff A contains all its cluster points, Der(A) Cl(A) = x X : ( N x)(N A = ) (20) { ∈ ∀ ∈ N 6 ∅ } A, iff A contains its closure. Hence ⊆ of all points in X adherent to A\is given by the A = Cl(A) Cl(A) union of A with its boundary. ⇔ The interior of A = x A : (( N x)(N A)) { ∈ ∃ ∈ N ⊆ def Int(A) = x X : ( N )(N A) (21) (( N x)(N (X A) = )) . { ∈ ∃ ∈ Nx ⊆ } ∨ ∀ ∈ N − 6 ∅ } \ 13By definition, an interval I in a totally ordered set X is a subset of X with the property (x , x I) (x X : x x x ) x I 1 2 ∈ ∧ 3 ∈ 1 ≺ 3 ≺ 2 ⇒ 3 ∈ so that any element of X lying between two elements of I also belongs to I. 14Although we do not pursue this point of view here, it is nonetheless tempting to speculate that the answer to the question “Why does the entropy of an isolated system increase?” may be found by exploiting this line of reasoning that seeks to explain the increase in terms of a visible component associated with the usual topology as against a different latent workplace topology that governs the dynamics of nature. December 3, 2003 12:13 00851

3164 A. Sengupta

Comparison of Eqs. (19) and (22) also makes it (g) Cl(A) = F X : F { ⊆ clear that Bdy(A) Der(A). The special case of ⊆ \is a closed set of X containing A A = Iso(A) with Der(A) X A is important } ⊆ − (25) enough to deserve a special mention: A straightforward consequence of property (b) Definition 2.4 (Donor set). A proper, nonempty is that the boundary of any subset A of a topolog- subset A of X such that Iso(A) = A with Der(A) ical space X is closed in X; this significant result ⊆ X A will be called self-isolated or donor. Thus se- may also be demonstrated as follows. If x X is not − ∈ quences eventually in a donor set converges only in the boundary of A there is some neighborhood in its complement; this is, the opposite of the N of x that does not intersect both A and X A. For each point y N, N is a neighborhood of −that characteristic of a closed set where all converging ∈ sequences eventually in the set must necessarily point that does not meet A and X A simultaneously so that N is contained wholly in− X Bdy(A). converge in it. A closed-donor set with a closed − neighbor has no derived or boundary sets, and will We may now take N to be open without any loss of be said to be isolated in X. generality implying thereby that X Bdy(A) is an open set of X from which it follows −that Bdy(A) is closed in X. Example 2.5. In an isolated set sequences con- Further material on topological spaces relevant verge, if they have to, simultaneously in the com- to our work can be found in Appendix A.3. plement (because it is donor) and in it (because it is closed). Convergent sequences in such a set can only End Tutorial 4 be constant sequences. Physically, if we consider adherents to be contributions made by the dynamics of the corresponding sequences, then an isolated set is Working in a general topological space, we now re- secluded from its neighbor in the sense that it nei- call the solution of an ill-posed problem f(x) = y ther receives any contributions from its surround- [Sengupta, 1997] that leads to a multifunctional in- ings, nor does it give away any. In this light and verse f − through the generalized inverse G. Let terminology, a closed set is a selfish set (recall that f: (X, ) (Y, ) be a (nonlinear) function be- U → V a set A is closed in X iff every convergent net of X tween two topological space (X, ) and (Y, ) that U V that is eventually in A converges in A; conversely a is neither one-one or onto. Since f is not one- set is open in X iff the only nets that converge in A one, X can be partitioned into disjoint equiva- are eventually in it), whereas a set with a derived lence classes with respect to the equivalence relation x x f(x ) = f(x ). Picking a representative set that intersects itself and its complement may be 1 ∼ 2 ⇔ 1 2 considered to be neutral. Appendix A.3 shows the member from each of the classes (this is possible various possibilities for the derived set and bound- by the Axiom of Choice; see the following Tuto- ary of a subset A of X. rial) produces a basic set XB of X; it is basic as it corresponds to the row space in the linear matrix Some useful properties of these concepts for example which is all that is needed for taking an a subset A of a topological space X are the inverse. XB is the counterpart of the quotient set X/ of Sec. 1, with the important difference that following. whereas∼ the points of the quotient set are the equiv- (a) Bdy (X) = , alence classes of X, XB is a subset of X with each X ∅ (b) Bdy(A) = Cl(A) Cl(X A), of the classes contributing a point to XB. It then − follows that f : X f(X) is the bijective re- (c) Int(A) = X Cl(X A) = A Bdy(A) = B B → − T − − striction a that reduces the original ill-posed Cl(A) Bdy(A), |row(A) − problem to a well-posed one with XB and f(X) (d) Int(A) Bdy(A) = , ∅ corresponding respectively to the row and column (e) X = Int(A) Bdy(A) Int(X A), −1 T − spaces of A, and fB : f(X) XB is the ba- → − (f) Int(A) = S G X :SG sic inverse from which the multiinverse f is ob- { ⊆ tained through G, which in turn corresponds to the [is an open set of X contained in A } Moore–Penrose inverse GMP. The topological con- (24) siderations (obviously not for inner product spaces December 3, 2003 12:13 00851

Toward a Theory of Chaos 3165

that applies to the Moore–Penrose inverse) needed of the choice of the single element π from the re- to complete the solution are discussed below and in als. To see this more closely in the context of maps Appendix A.1. that we are concerned with, let f : X Y be a non-injective, onto map. To construct a→functional right inverse f : Y X of f, we must choose, for r → each y Y one representative element xrep from Begin Tutorial 5: Axiom of Choice ∈− the set f (y) and define fr(y) to be that element and Zorn’s Lemma according to f f (y) = f(x ) = y. If there is ◦ r rep Since some of our basic arguments depend on it, no preferred or natural way to make this choice, this Tutorial contains a short description of the the axiom of choice allows us to make an arbitrary Axiom of Choice that has been described as “one selection from the infinitely many that may be pos- of the most important, and at the same time one sible from f −(y). When a natural choice is indeed of the most controversial, principles of mathemat- available, as for example in the case of the initial 0 ics”. What this axiom states is this: For any set X value problem y (x) = x; y(0) = α0 on [0, a], the 2 there exists a function fC : 0(X) X such that definite solution α0 +x /2 may be selected from the x f (A ) A for every non-emptP y subset→ A of X; infinitely many x0dx0 = α+x2/2, 0 x a that C α α α 0 ≤ ≤ here (∈X) is the class of all subsets of X except . are permissible, and the axiom of choice sanctions P0 ∅ R Thus, if X = x1, x2, x3 is a three element set, a this selection. In addition, each y Y gives rise to { } ∈ − possible choice function is given by the family of solution sets Ay = f (y) : y Y and the real power of the axiom is {its assertion∈that} f ( x , x , x ) = x , f ( x , x ) = x , C { 1 2 3} 3 C { 1 2} 1 it is possible to make a choice f (A ) A on every C y ∈ y fC( x2, x3 ) = x3, fC( x3, x1 ) = x3 , A simultaneously; this permits the choice on every { } { } y f ( x ) = x , f ( x ) = x , f ( x ) = x . Ay of the collection to be made at the same time. C { 1} 1 C { 2} 2 C { 3} 3 It must be appreciated that the axiom is only an ex- Pause Tutorial 5 istence result that asserts every set to have a choice function, even when nobody knows how to construct one in a specific case. Thus, for example, how does one pick out the isolated irrationals √2 or π from Figure shows our formulation and solution the uncountable reals? There is no doubt that they [Sengupta, 1997] of the inverse ill-posed problem do exist, for we can construct a right-angled trian- f(x) = y. In sub-diagram X X f(X), the surjec- − B− gle with sides of length 1 or a circle of radius 1. The tion p : X XB is the counterpart of the quotient axiom tells us that these choices are possible even map Q of Fig.→ 2 that is known in the present con- though we do not know how exactly to do it; all text as the identification of X with XB (as it iden- that can be stated with confidence is that we can tifies each saturated subset of X with its represen- actually pick up rationals arbitrarily close to these tative point in XB), with the space (XB, FT ; p ) irrationals. carrying the identification topology FT ; p {Ubeing} The axiom of choice is essentially meaningful known as an identification space. By sub-diagram{U } when X is infinite as illustrated in the last two ex- Y XB f(X), the image f(X) of f gets the amples. This is so because even when X is denu- subsp− ace top− ology15 IT j; from (Y, ) by the in- merable, it would be physically impossible to make clusion j : f(X) {Y Vwhen} its opVen sets are an infinite number of selections either all at a time generated as, and →only as, j−1(V ) = V f(X) or sequentially: the Axiom of Choice nevertheless for V . Furthermore if the bijection fB con- ∈ V T tells us that this is possible. The real strength and necting XB and f(X) (which therefore acts as a utility of the Axiom however is when X and some 1 : 1 correspondence between their points, imply- or all of its subsets are uncountable as in the case ing that these sets are set-theoretically identical

15 In a subspace A of X, a subset UA of A is open iﬀ UA = A U for some open set U of X. The notion of subspace topology can be formalized with the help of the inclusion map i : A (X, ) that puts every point of A back to where it came from, → U thus T

A = UA = A U : U U { ∈ U} = i−(U) : U . { T∈ U} December 3, 2003 12:13 00851

3166 A. Sengupta

indistinguishable which may be considered to be

¦© ¦¨§ ©

¢¡ identical in as far as their topological properties

are concerned.

#, ,"

Remark. It may be of some interest here to spec-

+ * ulate on the signiﬁcance of ininality in our work.

Physically, a map f: (X, ) (Y, ) between two spaces can be taken to represenU → t anVinteraction be-

tween them and the algebraic and topological char-

¦ © "!# ¦ ¦$ © %&'( )!#

acters of f determine the nature of this interaction.

¡¤£

¥ A simple bijection merely sets up a correspondence, that is an interaction, between every member of X with some member Y , whereas a continuous map Fig. 5. Solution of ill-posed problem f(x) = y, f : X Y . → establishes the correspondence among the special G : Y X , a generalized inverse of f because of fGf = f → B category of “open” sets. Open sets, as we see in and GfG = G which follows from the commutativity of the Appendix A.1, are the basic ingredients in the the- diagrams, is a functional selection of the multi-inverse f −: ory of convergence of sequences, nets and ﬁlters, and (Y, ) – (X, )

tion such as fB is known as a homeomorphism FT ; f< = IT j; and ininality for functions that are neither 1 : 1 {U } { V} (26) IT f; = FT ; p ; nor onto is a generalization of homeomorphism for {< V} {U }} bijections; refer Eqs. (A.47) and (A.48) for a set- in what follows we will refer to the injective and sur- theoretic formulation of this distinction. A homeo- jective restrictions of f by their generic topological morphism f: (X, ) (Y, ) renders the home- symbols of embedding e and association q, respec- omorphic spaces (UX, →) andV (Y, ) topologically tively. What are the topological characteristics of f U V 16A surjective function is an association iﬀ it is image continuous and an injective function is an embedding iﬀ it is preimage continuous. December 3, 2003 12:13 00851

Toward a Theory of Chaos 3167

in order that the requirements of Eq. (26) be met? £ From Appendix A.1, it should be clear by super- posing the two parts of Fig. 21 over each other that ¡ given q : (X, ) (f(X), FT ; q ) in the ﬁrst of ¤ U → {U }

these equations, IT j; will equal FT ; q iﬀ j ¥ { V} {U } is an ininal open inclusion and Y receives FT ; f . ¦ In a similar manner, preimage continuity of{Ue re-} quires p to be open ininal and f to be preimage continuous if the second of Eq. (26) is to be satisﬁed.

Thus under the restrictions imposed by Eq. (26),

¥ ¡ § £

the interaction f between X and Y must be such

¤ ¢ as to give X the smallest possible topology of f- ¢ saturated sets and Y the largest possible topology of images of all these sets: f, under these condi- 2x, 0 x < 3/8 tions, is an ininal transformation. Observe that a ≤ Fig. 6. The function f(x) = 3/4, 3/8 x 5/8 direct application of parts (b) of Theorems A.2.1  ≤ ≤  7/6 2x/3, 5/8 < x 1. and A.2.2 to Fig. implies that Eq. (26) is satisfied − ≤ iff fB is ininal, that is iff it is a homeomorphism.  Ininality of f is simply a reflection of this as it is neither 1 : 1 nor onto. An injective branch of a function f in this work The f- and p-images of each saturated set of X refers to the restrictions fB and its associated in- −1 are singletons in Y (these saturated sets in X arose, verse fB . in the first place, as f −( y ) for y Y ) and in X , The following example of an inverse ill-posed { } ∈ B respectively. This permits the embedding e = j f problem will be useful in fixing the notations intro- ◦ B to give XB the character of a virtual subspace of Y duced above. Let f on [0, 1] be the function shown just as i makes f(X) a real subspace. Hence the in- below. verse images p−(x ) = f −(e(x )) with x X , and Then f(x) = y is well-posed for [0, 1/4), and ill- r r r ∈ B q−(y) = f −(i(y)) with y = f (x ) f(X) are the posed in [1/4, 1]. There are two injective branches B r ∈ same, and are just the corresponding f − images via of f in [1/4, 3/8) (5/8, 1] , and f is constant the injections e and i, respectively. G, a left inverse ill-posed {in [3/8, 5/8]. Hence}the basic component S of e, is a generalized inverse of f. G is a general- fB of f can be taken to be fB(x) = 2x for x −1 ∈ ized inverse because the two set-theoretic defining [0, 3/8) having the inverse fB (y) = x/2 with requirements of fGf = f and GfG = G for the y [0, 3/4]. The generalized inverse is obtained generalized inverse are satisfied, as Fig. shows, in by∈taking [0, 3/4] as a subspace of [0, 1], while the the following forms multiinverse f − follows by associating with every point of the basic domain [0, 1]B = [0, 3/8], the re- jfBGf = f GjfBG = G . spective equivalent points [3/8]f = [3/8, 5/8] and [x]f = x, 7/4 3x forx [1/4, 3/8). Thus the in- In fact the commutativity embodied in these equali- verses G{ and f−− of}f are17∈ ties is self evident from the fact that e = ifB is a left inverse of G, that is eG = 1 . On putting back X Y B y 3 into X by identifying each point of XB with the set , y 0, 2 ∈ 4 it came from yields the required set-valued inverse G(y) =  , f −, and G may be viewed as a functional selection  3 −  0, y , 1 of the multiinverse f . ∈ 4  17If y / (f) then f −( y ) := which is true for any subset of Y (f). However from the set-theoretic definition of ∈ R { } ∅ − R natural numbers that requires 0 := , 1 = 0 , 2 = 0, 1 to be defined recursively, it follows that f −(y) can be identified ∅ { } { } with 0 whenever y is not in the domain of f −. Formally, the successor set A+ = A A of A can be used to write 0 := , { } ∅ 1 = 0+ = 0 0 , 2 = 1+ = 1 1 = 0 1 3 = 2+ = 2 2 = 0 1 2 , etc. Then the set of natural numbers N is { } { } { } { } { } { } { } { } defined to be the intersection of all the successor sets, where a successor set is anySset that contains and A+ whenever A S ∅ belongs to S. Observe how in theS successorSnotation, countableS union ofSsingletonS integers recursively define the corresponding S sum of integers. December 3, 2003 12:13 00851

3168 A. Sengupta

y 1 is the unique matrix representation of the functional , y 0, −1 5 2 ∈ 2 inverse aB : a(R ) XB extended to Y defined  according to18 →  y 7 3y 1 3  , , y , −1  2 4 − 2 ∈ 2 4 a (b), if b (a) −  g(b) = B ∈ R (29) f (y) =   3 5 3 0, if b Y (a) ,  , , y = ∈ − R 8 8 4 that bears comparison with the basic inverse   3 5 1  0, y , 1 , 0 0  ∈ 4 2 −2     0 0 0 0 which shows that f − is multivalued. In order to −1 ∗  3 1  avoid cumbersome notations, an injective branch of AB (b ) =    0 0  f will always refer to a representative basic branch  −4 4    f , and its “inverse” will mean either f −1 or G.  0 0 0 0  B B    0 0 0 0    Example 2.3 (Revisited). The row reduced   echelon form of the augmented matrix (A b) of b1 | Example 2.3 is b2 5   : a(R ) XB × 2b1 → 3 1 5b1 b2   1 3 0  b2 b1  − 2 2 2 − 2  −    between the two-dimensional  column and row 1 3 3b1 b2 (A b)  0 0 1 +  (27) spaces of A which is responsible for the particular | →  −4 4 − 4 4  −1   solution of Ax = b. Thus G is simply AB acting  0 0 0 0 0 2b1 + b3   −  on its domain a(X) considered a subspace of Y ,  0 0 0 0 0 b1 b2 + b4  suitably extended to the whole of Y . That it is in-  −    deed a generalized inverse is readily seen through − The multifunctional solution x = A b, with b any the matrix multiplications GAG and AGA that 4 element of Y = R not necessarily in the image of can be verified to reproduce G and A, respectively. a, is Comparison of Eqs. (12) and (29) shows that the 3 1 Moore–Penrose inverse differs from ours through the geometrical constraints imposed in its defini- 3 −2 −2     tion, Eqs. (13). Of course, this results in a more 1 0 0   complex inverse (14) as compared to our very simple −  1   3  x=A b=Gb+x2 0 +x4  +x5   , (28); nevertheless it is true that both the inverses        0   4   −4  satisfy        0   1   0        1 0 0 0 0    0   1      T 0 1 0 0 0     E((E(GMP)) ) =   with its multifunctional character arising from the 0 0 0 0 0   arbitrariness of the coefficients x2, x4 and x5. The  0 0 0 0 0    generalized inverse   = E((E(G))T) 5 1 0 0 where E(A) is the row-reduced echelon form of A. 2 −2   The canonical simplicity of Eq. (28) as compared to 0 0 0 0 Eq. (14) is a general feature that suggests a more  3 1  G =   : Y XB (28) natural choice of bases by the map a than the or-  0 0  →  −4 4  thogonal set imposed by Moore and Penrose. This    0 0 0 0  is to be expected since the MP inverse, governed by    0 0 0 0  Eq. (13), is a subset of our less restricted inverse     18See footnote 17 for a justification of the definition when b is not in (a). R December 3, 2003 12:13 00851

Toward a Theory of Chaos 3169

described by only the first two of (13); more specifi- the basis that diagonalizes an n n matrix (when cally the difference is made clear in Fig. 4(a) which this is possible) is not the standard× “diagonal” or- n shows that for any b / (A), only GMP(b⊥) = 0 thonormal basis of R , but a problem-dependent, as compared to G(b) ∈=R0. This seems to imply less canonical, basis consisting of the n eigenvectors that introducing extraneous topological considera- of the matrix. The 0-rows of the inverse of Eq. (28) tions into the purely set-theoretic inversion process result from the three-dimensional null-space vari- may not be a recommended way of inverting, and ables x2, x4 and x5, while the 0-columns come from the simple bases comprising the row and null spaces the two-dimensional image-space dependency of b3, T of A and A — that are mutually orthogonal just as b4 on b1 and b2, that is from the last two zero rows those of the Moore–Penrose — are a better choice of the reduced echelon form (27) of the augmented for the particular problem Ax = b than the gen- matrix. eral orthonormal bases that the MP inverse intro- We will return to this theme of the generation duces. These “good” bases, with respect to which of a most appropriate problem-dependent topology the generalized inverse G has a considerably sim- for a given space in the more general context of pler representation, are obtained in a straightfor- chaos in Sec. 4.2. ward manner from the row-reduced forms of A and In concluding this introduction to generalized AT. These bases are inverses we note that the inverse G of f comes very close to being a right inverse: thus even though (a) The column space of A is spanned by the AG = 12 its row-reduced form columns (1, 3, 2, 2)T and (1, 5, 2, 4)T of A that 66 correspond to the basic columns containing the 1 0 0 0 0 1 0 0 leading 1’s in the row-reduced form of A,   (b) The null space of AT is spanned by the solu- 0 0 0 0 T T   tions ( 2, 0, 1, 0) and (1, 1, 0, 1) of the  0 0 0 0  − T −   equation A b = 0,   (c) The row space of A is spanned by the rows is to be compared with the corresponding less (1, 3, 2, 1, 2) and (3, 9, 10, 2, 9) of A cor- satisfactory − − responding to the non-zero rows in the row- 1 0 2 1 − reduced form of A, 0 1 0 1 (d) The null space of A is spanned by the   0 0 0 0 solutions (3, 1, 0, 0, 0), ( 6, 0, 1, 4, 0), and   −  0 0 0 0  ( 2, 0, 3, 0, 4) of the equation Ax = 0.   − −   representation of AGMP. The main differences between the natural “good” bases and the MP-bases that are responsible for the difference in the form of inverses, is 3. Multifunctional Extension of that the latter have the additional restrictions of Function Spaces being orthogonal to each other (recall the orthog- The previous section has considered the solution of onality property of the Q-matrices), and the more ill-posed problems as multifunctions and has shown severe of basis vectors mapping onto basis vectors how this solution may be constructed. Here we in- according to Axi = σibi, i = 1, . . . , r, where the troduce the multifunction space Multi|(X) as the n m T xi i=1 and bj j=1 are the eigenvectors of A A first step toward obtaining a smallest dense ex- { } T { } r and AA respectively and (σi)i=1 are the positive tension Multi(X) of the function space Map(X). T square roots of the non-zero eigenvalues of A A (or Multi|(X) is basic to our theory of chaos [Sengupta of AAT), with r denoting the dimension of the row & Ray, 2000] in the sense that a chaotic state of or column space. This is considered as a serious re- a system can be fully described by such an inde- striction as the linear combination of the basis b terminate multifunctional state. In fact, multifunc- { j} that Axi should otherwise have been equal to, al- tions also enter in a natural way in describing the lows a greater flexibility in the matrix representa- spectrum of nonlinear functions that we consider in tion of the inverse that shows up in the structure of Sec. 6; this is required to complete the construc- G. These are, in fact, quite general considerations in tion of the smallest extension Multi(X) of the func- the matrix representation of linear operators; thus tion space Map(X). The main tool in obtaining the December 3, 2003 12:13 00851

3170 A. Sengupta

space Multi|(X) from Map(X) is a generalization of (fα)α∈D converge pointwise in Y . Explicitly, this of the technique of pointwise convergence of con- is the subset of Y on which subnets of injective tinuous functions to (discontinuous) functions. In branches of (fα)α∈D in Map(Y, X) combine to form the analysis below, we consider nets instead of se- a net of functions that converge pointwise to a fam- quences as the spaces concerned, like the topology ily of limit functions G : X. Depending on R− → of pointwise convergence, may not be first count- the nature of (fα)α∈D, there may be more than one able, Appendix A.1. − with a corresponding family of limit functions onR each of them. To simplify the notation, we will usually let G : X denote all the limit func- 3.1. Graphical convergence of a net R− → tions on all the sets −. of functions If we consider cofinalR rather than residual sub- Let (X, ) and (Y, ) be Hausdorff spaces and sets of D then corresponding D and R can be U V + + (fα)α∈D : X Y be a net of piecewise continuous expressed as functions, not→necessarily with the same domain or range, and suppose that for each α D there is + = x X : ((fν (x))ν∈Cof(D) converges in ∈ − D { ∈ a finite set Iα = 1, 2, . . . Pα such that fα has Pα (Y, )) (32) functional branches{ possibly with} different domains; V } + = y Y : ( i Iν)((gνi(y))ν∈Cof(D) obviously Iα is a singleton iff f is a injective. For R { ∈ ∃ ∈ D converges in (X, )) . (33) each α , define functions (gαi)i∈Iα : Y X U } such that∈ → It is to be noted that the conditions + = − and I D D fαgαifα = fαi i = 1, 2, . . . Pα , + = − are necessary and sufficient for the Kura- RtowskiRconvergence to exist. Since and differ where f I is a basic injective branch of f on + + αi α from and only in having cofinalD subsetsR of D some subset of its domain: g f I = 1 on (f I ), − − αi αi X αi replacedD by residualR ones, and since residual sets are f I g = 1 on (g ) for each i I . TheDuse of αi αi Y αi α also cofinal, it follows that and . nets and filters isD dictated by the∈fact that we do − + − + The sets and servDe for⊆ theD convRergence⊆ Rof not assume X and Y to be first countable. In the − − a net of functionsD Rjust as and are for the application to the theory of dynamical systems that + + convergence of subnets of theD nets (adherR ence). The follows, X and Y are compact subsets of R when the latter sets are needed when subsequences are to be use of sequences suffice. considered as sequences in their own right as, for In terms of the residual and cofinal subsets example, in dynamical systems theory in the case Res(D) and Cof(D) of a directed set D (Defini- of ω-limit sets. tion A.1.7), with x and y in the equations below As an illustration of these definitions, consider being taken to belong to the required domains, de- the sequence of injective functions on the interval fine subsets − of X and − of Y as n n D R [0, 1] fn(x) = 2 x, for x [0, 1/2 ], n = 0, 1, 2 . . . . ∈ − = x X : ((fν (x))ν∈D converges in (Y, )) Then D0.2 is the set 0, 1, 2 and only D0 is even- D { ∈ V } tual in D. Hence {is the single} point set 0 . On (30) D− { } the other hand Dy is eventual in D for all y and − − = y Y : ( i Iν)((gνi(y))ν∈D converges in R R { ∈ ∃ ∈ is [0, 1]. (X, )) (31) U } Thus, Definition 3.1 (Graphical Convergence of a net of is the set of points of X on which the values functions). A net of functions (fα)α∈D : (X, ) D− U → of a given net of functions (fα)α∈D converge point- (Y, ) is said to converge graphically if either − = orV = ; in this case let F : YD and6 wise in Y . Explicitly, this is the subset of X on ∅ R− 6 ∅ D− → 19 G : X be the entire collection of limit func- which subnets in Map(X, Y ) combine to form a R− → net of functions that converge pointwise to a limit tions. Because of the assumed Hausdorffness of X function F : − Y . and Y , these limits are well defined. is theDset→of points of Y on which the values The graph of the graphical limit of the net − MG of theRnets in X generated by the injective branches (f ) : (X, ) (Y, ) denoted by f , is the α U → V α → M

19A subnet is the generalized uncountable equivalent of a subsequence; for the technical deﬁnition, see Appendix A.1. December 3, 2003 12:13 00851

Toward a Theory of Chaos 3171

subset of − − that is the union of the graphs spaces X and Y to be a consequence of both the di- of the functionD ×FR and the multifunction G− rect interaction represented by f : X Y and also − → − the inverse interaction f : Y – X, and our formu- GM = GF GG → lation of pointwise biconvergence→ is a formalization where [ of this idea. Thus the basic examples (1) and (2) GG− = (x, y) X Y : (y, x) GG Y X . below produce multifunctions instead of discontin- { ∈ × ∈ ⊆ × } uous functions that would be obtained by the usual pointwise limit. Begin Tutorial 6: Graphical Convergence Example 3.1 The following two examples are basic to the understanding of the graphical convergence of func- (1) 0, 1 x 0 tions to multifunctions and were the examples − ≤ ≤ that motivated our search of an acceptable tech-  1 nx, 0 x nique that did not require vertical portions of fn(x) =  : [ 1, 1] [0, 1]  ≤ ≤ n − → limit relations to disappear simply because they  1 1, x 1 were non-functions: the disturbing question that n ≤ ≤  needed an answer was how not to mathemati-  cally sacriﬁce these extremely signiﬁcant physi-  y 1 g (y) = : [0, 1] 0, cal components of the limiting correspondences. n n → n Furthermore, it appears to be quite plausible to expect a physical interaction between two Then

0, 1 x 0 F (x) = − ≤ ≤ on = = [ 1, 0] (0, 1] 1, 0 < x 1 D− D+ − ≤ S G(y) = 0 on = [0, 1] = . R− R+

The graphical limit is ([ 1, 0], 0) (0, [0, 1]) not converge graphically because in this case both − ((0, 1], 1). the sets − and − are empty. The power of S S D R (2) fn(x) = nx for x [0, 1/n] gives gn(y) = y/n : graphical convergence in capturing multifunctional [0, 1] [0, 1/n]. Then∈ limits is further demonstrated by the example of → ∞ the sequence (sin nπx)n=1 that converges to 0 both 1-integrally and test-functionally, Eqs. (3) and (4). F (x) = 0 on − = 0 = + , D { } D It is necessary to understand how the concepts G(y) = 0 on = [0, 1] = . R− R+ of eventually in and frequently in of Appendix A.2 apply in examples (a) and (b) of Fig. 7. In these The graphical limit is (0, [0, 1]). two examples we have two subsequences one each In these examples that we consider to be the proto- for the even indices and the other for the odd. types of graphical convergence of functions to mul- For a point-to-point functional relation, this would tifunctions, G(y) = 0 on because g (y) 0 mean that the sequence frequents the adherence set R− n → for all y −. Compare the graphical multifunc- adh(x) of the sequence (xn) but does not converge tional limits∈ Rwith the corresponding usual pointwise anywhere as it is not eventually in every neigh- functional limits characterized by discontinuity at borhood of any point. For a multifunctional limit x = 0. Two more examples from Sengupta and Ray however it is possible, as demonstrated by these [2000] that illustrate this new convergence princi- examples, for the subsequences to be eventually ple tailored speciﬁcally to capture one-to-many re- in every neighborhood of certain subsets common lations are shown in Fig. 7 which also provides an to the eventual limiting sets of the subsequences; example in Fig. 7(c) of a function whose iterates do this intersection of the subsequential limits is now December 3, 2003 12:13 00851

3172 A. Sengupta

1 2 − 1 1+11+11+1/n1+1/n/n /n1/n1/n1/n2 −2/n1−2/n−1/n1/n 1.5 1.51.5 /n n eveneveneven 1.5 even n evn enn nn evnenevn eneven 11 1 1 1+21+/n 2 1+2/n1+/n2/n 3+1 3+3+/n13+1/n1/n 0 0 0 /n 0 12 123 121212 123123123 -0.5 n odnd odn odd d -0.5-0.5-0.5 n odd noddododd d nodndn (a) (b) (a)(a) (a)(a) (b)(b)(b)(b)

1 1 1 1 − − 2 2 2 2 2 2 12 iter12ates12iteriteratesofatesof0.05of−0+−.050x.05+ x+−2x −x x 12 iter1212atesiteriteratesofates0.of7+of0.x7+0.+7+x+x2x+x 12 iterates of −0.05 + x − x 1 1 12 iterates of 0.7+x+x 1 1 0 0 0 α α 0 α α

1 1 1 -1 -1-1 1 -1 0 0 0 0 12 1212 12 -2 -2-2 -2 -1 -1 -1 -1 -3 -3-3 -3-1 -1 -1 0 0 0 1 1 1 2 2 2 acacac -1 0 1 2 ac (c) (c)(c) (d) (d)(d) (c)(c) (d) (d) 1 for 0 x 1 Fig. 7. The graphical limits are: (a) F (x) = ≤ ≤ on − = [0, 1] (1, 2], and G(y) = 1 on − = [0, 1]. Also 0 for 1 < x 2 D R ≤ 1 on + = [0, 3/2] S G = R . 1 on + = [ 1/2, 1] R − (b) F (x) = 1 on − = 0 and G(y) = 0 on − = 1 . Also F (x) = 1/2, 0, 1, 3/2 respectively on + = (0, 3], 2 , D { } R { } − D { } 0 , (0, 2) and G(y) = 0, 0, 2, 3 respectively on + = ( 1/2, 1], [1, 3/2), [0, 3/2), [ 1/2, 0). { } 2 R − − (c) For f(x) = 0.05 + x x , no graphical limit as − = = −. − −2 D ∅ R (d) For f(x) = 0.7 + x x , F (x) = α on − = [a, c], G (y) = a and G (y) = c on − = ( , α]. Notice how the two ﬁxed − D 1 2 R −∞ points and their equivalent images deﬁne the converged limit rectangular multi. As in example (1) one has − = +; also D D − = +. R R

defined to be the limit of the original sequence. A ous set of equations (sequence) may have distinct similar situation obtains, for example, in the solu- solutions (limits), the solution of the equations is tion of simultaneous equations: The solution of the their common point of intersection. equation a11x1 + a12x2 = b1 for one of the1 vari-1 Considered as sets in X Y , the discussion of 1 × ables x2 say with a12 = 0, is the set represen1 ted convergence of a sequence of graphs f : X Y 6 n → by the straight line x2 = m1x1 + c1 for all x1 in would be incomplete without a mention of the con- its domain, while for a different set of constants vergence of a sequence of sets under the Hausdorff a21, a22 and b2 the solution is the entirely differ- metric that is so basic in the study of fractals. In ent set x2 = m2x1 + c2, under the assumption that this case, one talks about the convergence of a se- m = m and c = c . Thus even though the indi- quence of compact subsets of the metric space Rn 1 6 2 1 6 2 vidual equations (subsequences) of the simultane- so that the sequences, as also the limit points that December 3, 2003 12:13 00851

Toward a Theory of Chaos 3173

are the fractals, are compact subsets of Rn. Let topology of pointwise convergence iff (f ) converges K α denote the collection of all nonempty compact sub- pointwise to f in the sense that fα(x) f(x) in Y n → sets of R . Then the Hausdorff metric dH between for every x in X. two sets on is defined to be Proof. Necessity. First consider f f in K α → dH(E, F ) = max δ(E, F ), δ(F, E) E, F , (Map(X, Y ), ). For an open neighborhood V of { } ∈ K f(x) in Y withTx X, let B(x; V ) be a local neigh- where borhood of f in (Map∈ (X, Y ), ), see Eq. (A.6) in T δ(E, F ) = max min x y 2 Appendix A.1. By assumption of convergence, (fα) x∈E y∈F k − k must eventually be in B(x; V ) implying that fα(x) is δ(E, F ) is the non-symmetric 2-norm in Rn. is eventually in V . Hence f (x) f(x) in Y . α → The power and utility of the Hausdorff distance is Sufficiency. Conversely, if fα(x) f(x) in best understood in terms of the dilations E + ε := Y for every x X, then for a finite→collection n I ∈ x∈E Dε(x) of a subset E of R by ε where Dε(x) of points (xi)i=1 of X (X may itself be uncount- I is a closed ball of radius ε at x; physically a dilation able) and corresponding open sets (Vi)i=1 in Y with Sof E by ε is a closed ε-neighborhood of E. Then a f(x ) V , let B((x )I ; (V )I ) be an open neigh- i ∈ i i i=1 i i=1 fundamental property of dH is that dH(E, F ) ε borhood of f. From the assumed pointwise conver- iff both E F + ε and F E + ε hold simultane-≤ gence f (x ) f(x ) in Y for i = 1, 2, . . . , I, it ⊆ ⊆ α i → i ously which leads [Falconer, 1990] to the interesting follows that (fα(xi)) is eventually in Vi for every I consequence that (xi)i=1. Because D is a directed set, the existence of ∞ If (Fn)n=1 and F are nonempty compact sets, a residual applicable globally for all i = 1, 2, . . . , I then lim F = F in the Hausdorff metric iff is assured leading to the conclusion that f (x ) V n→∞ n α i ∈ i Fn F +ε and F Fn +ε eventually. Furthermore eventually for every i = 1, 2, . . . , I. Hence fα ⊆ ∞ ⊆ I I ∈ if (Fn)n=1 is a decreasing sequence of elements of a B((xi)i=1; (Vi)i=1) eventually; this completes the n filter-base in R , then the nonempty and compact demonstration that fα f in (Map(X, Y ), ), limit set F is given by and thus of the proof. → T ∞ End Tutorial 6 lim Fn = F = Fn . n→∞ n\=1 Note that since Rn is Hausdorff, the assumed compactness of Fn ensures that they are also closed in 3.2. The extension Multi|(X, Y ) of Rn; F , therefore, is just the adherent set of the Map (X, Y ) filter-base. In the deterministic algorithm for the In this section we show how the topological treat- generation of fractals by the so-called iterated func- ment of pointwise convergence of functions to function system (IFS) approach, Fn is the inverse im- tions given in Example A.1.1 of Appendix 1 can be age by the nth iterate of a non-injective function f generalized to generate the boundary Multi|(X, Y ) having a finite number of injective branches and between Map(X, Y ) and Multi(X, Y ); here X converging graphically to a multifunction. Under and Y are Hausdorff spaces and Map(X, Y ) and the conditions stated above, the Hausdorff metric Multi(X, Y ) are respectively the sets of all func- ensures convergence of any class of compact sub- tional and non-functional relations between X and Rn sets in . It appears eminently plausible that our Y . The generalization we seek defines neighbor- Rn multifunctional graphical convergence on Map( ) hoods of f Map(X, Y ) to consist of those func- Rn implies Hausdorff convergence on : in fact point- tional relations∈ in Multi(X, Y ) whose images at any wise biconvergence involves simultaneous conver- point x X lies not only arbitrarily close to f(x) gence of image and preimage nets on Y and X, (this generates∈ the usual topology of pointwise con- respectively. Thus confining ourselves to the sim- vergence Y of Example A.1.1) but whose inverse pler case of pointwise convergence, if (fα)α∈D is a images atTy = f(x) Y contain points arbitrar- net of functions in Map(X, Y ), then the following ily close to x. Thus the∈ graph of f must not only theorem expresses the link between convergence in lie close enough to f(x) at x in V , but must addi- Map(X, Y ) and in Y . tionally be such that f −(y) has at least branch in Theorem 3.1. A net of functions (fα)α∈D con- U about x; thus f is constrained to cling to f as verges to a function f in (Map(X, Y ), ) in the the number of points on the graph of f increases T December 3, 2003 12:13 00851

3174 A. Sengupta

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Fig. 8. The power of graphical convergence, illustrated for Example 3.1 (1), shows a local neighborhood of the functions 4 4 4 x and 2x in (a) and (b) at the four points (xi)i=1 with corresponding neighborhoods (Ui)i=1 and (Vi)i=1 at (xi, f(xi)) in R in the X and Y directions respectively, see Eqs. (34) and (A.6) for the notations. (a) shows a function g in a pointwise neighborhood of f determined by the open sets Vi, while (b) shows g in a graphical neighborhood of f due to both Ui and Vi. A comparison of these ﬁgures demonstrates how the graphical neighborhood forces functions close to f remain closer to it than if they were in its pointwise neighborhood. This property is clearly visible in (a) where g, if it were to be in a graphical neighborhood of f, would be more faithful to it by having to be also in U2 and U4. Thus in this case not only must the images j j f(xij ) f(xi) as Vi decreases, but also the preimages xij xi with shrinking Ui. It is this simultaneous convergence of → → both images and preimages at every x that makes graphical convergence a natural candidate for multifunctional convergence of functions.

with convergence and, unlike in the situation of sim- for every choice of α D, is a base T of ple pointwise convergence, no gaps in the graph of (Map(X, Y ), ). Here the∈ directed set D is Bused the limit object is permitted not only, as in Exam- as an indexingT tool because, as pointed out in Ex- ple A.1.1 on the domain of f, but simultaneously ample A.1.1, the topology of pointwise convergence on its range too. We call the resulting generated is not ﬁrst countable. topology the topology of pointwise biconvergence on In a manner similar to Eq. (34), the open sets Map(X, Y ), to be denoted by . Thus for any given of (Multi(X, Y ), ˆ ), where Multi(X, Y ) are mul- T integer I 1, the generalization of Eq. (A.6) gives tifunctions with onlyT countably many values in Y ≥ for i = 1, 2, . . . , I, the open sets of (Map(X, Y ), ) for every point of X (so that we exclude continuous T to be regions from our discussion except for the “vertical lines” of Multi|(X, Y )), can be deﬁned as B((xi), (Vi); (yi), (Ui))

= g Map(X, Y ) : (g(x ) V ) Bˆ((xi), (Vi); (yi), (Ui)) { ∈ i ∈ i (g−(y ) U = ), i = 1, 2, . . . , I , (34) = Multi(X, Y ) : ( (xi) Vi = ) ∧ i i 6 ∅ } {G ∈ G 6 ∅ − \ I \I ( (yi) Ui = ) , (36) where (xi)i=1, (Vi)i=1 are as in that example, ∧ G 6 ∅ } I (yi)i=1 Y , and the corresponding open sets where \ I ∈ 20 (Ui) in X are chosen arbitrarily. A local base i=1 − at f, for (x , y ) G , is the set of functions (y) = x X : y (x) . i i ∈ f G { ∈ ∈ G } of (34) with yi = f(xi) and the collection of all I I I and (xi)i=1 ( ), (Vi)i=1; (yi)i=1 ( ), local bases I ∈ D M ∈ R M (Ui)i=1 are chosen as in the above. The topology B = B((x )Iα , (V )Iα ; (y )Iα , (U )Iα ) , (35) ˆ of Multi(X, Y ) is generated by the collection of α i i=1 i i=1 i i=1 i i=1 T

20Equation (34) is essentially the intersection of the pointwise topologies (A.6) due to f and f −. December 3, 2003 12:13 00851

Toward a Theory of Chaos 3175

all local bases Bˆ for every choice of α D, and it is persets of all elements of F ; see Appendix A.1) and α ∈ B not difficult to see from Eqs. (34) and (36), that the thereby the filter-base restriction ˆ of ˆ to Map(X, Y ) is just . ˆ ˆ T |Map(X, Y ) T T F = B = B mˆ : B F Henceforth ˆ and will be denoted by the B { { } ∈ B} T T ˆ [ same symbol , and convergence in the topology on M; this filter-base at m can also be obtained T ˆ of pointwise biconvergence in (Multi(X, Y ), ) will independently from Eq. (36). Obviously F is an T ˆ B be denoted by ⇒, with the notation being derived extension of F on M and F is the filter induced ˆ B B from Theorem 3.1. on M by F . We may also consider the filter-base to be a topologicalB base on M that defines a coarser topology on M (through all unions of members Definition 3.2 (Functionization of a multifunction). T of F ) and hence the topology A net of functions (fα)α∈D in Map(X, Y ) converges B in (Multi(X, Y ), ), fα ⇒ , if it biconverges ˆ = Gˆ = G mˆ : G T M pointwise in (Map(X, Y ), ∗). Such a net of func- T { { } ∈ T } T [ tions will be said to be a functionization of . on Mˆ to be the topology associated with ˆ. A finer M topology on Mˆ may be obtained by addingF to ˆ T Theorem 3.2. Let (f ) D be a net of functions in all the discarded elements of that do not satisfy α α∈ T0 Map(X, Y ). Then FIP. It is clear that mˆ is on the boundary of M

G because every neighborhood of mˆ intersects M by f f ⇒ . construction; thus (M, ) is dense in (Mˆ, ˆ ) which α → M ⇔ α M is the required topologicalT extension of (MT, ). T Proof. If (f ) converges graphically to then ei- In the present case, a filter-base at f α M ∈ ther or is non-empty; let us assume both of Map(X, Y ) is the neighborhood system F f at f D− R− B them to be so. Then the sequence of functions (fα) given by decreasing sequences of neighborhoods converges pointwise to a function F on − and to (Vk) and (Uk) of f(x) and x, respectively, and the D ˆ functions G on −, and the local basic neighbor- filter is the neighborhood filter f G where R G MultiF (X, Y ). We shall presentNan alternate, hoods of F and G generate the topology of point- ∈ | S wise biconvergence. and perhaps more intuitively appealing, description Conversely, for pointwise biconvergence on X of graphical convergence based on the adherence set of a filter in Sec. 4.1. and Y , − and − must be non-empty. R D As more serious examples of the graphical convergence of a net of functions to multifunction than Observe that the boundary of Map(X, Y ) in those considered above, Fig. 9 shows the first four the topology of pointwise biconvergence is a “line iterates of the tent map parallel to the Y -axis”. We denote this closure of 1 Map(X, Y ) as 2x, 0 x < ≤ 2 t(x) =  (t1 = t) . Definition 3.3. Multi ((X, Y ), ) = Cl(Map((X, 1 | T  2(1 x), x 1 Y ), )).  − 2 ≤ ≤ T defined on [0.1] and the sine map f = The sense in which Multi (X, Y ) is the smallest  n | sin(2n−1πx) , n = 1, . . . , 4 with domain [0, 1]. closed topological extension of M = Map(X, Y ) is | These examples| illustrate the important gener- the following, refer to Theorems A.1.4 and its proof. alization that periodic points may be replaced by the Let (M, ) be a topological space and suppose that T0 more general equivalence classes where a sequence of functions converges graphically; this generaliza- Mˆ = M mˆ { } tion based on the ill-posed interpretation of dynam- is obtained by adjoining an[extra point to M; here ical systems is significant for non-iterative systems M = Map(X, Y ) and mˆ Cl(M) is the multifunc- as in second example above. The equivalence classes ˆ ∈ of the tent map for its two fixed points 0 and 2/3 tional limit in M = Multi|(X, Y ). Treat all open sets of M generated by local bases of the type (35) generated by the first four iterates are with finite intersection property as a filter-base F 1 1 3 1 5 3 7 B [0]4 = 0, , , , , , , , 1 on X that induces a filter on M (by forming su- 8 4 8 2 8 4 8 F December 3, 2003 12:13 00851

3176 A. Sengupta

1111

0 0First 4Firiterstates4 iterofatestentofmatenp t map1 0 1 0Graph Grof apﬁrhsto4f |ﬁsinerst |4ma|sineps| maps11

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Fig. 9. The ﬁrst four iterates of (a) tent and (b) sin(2n−1πx) maps show the formal similarity of the dynamics of these | | ∞ functions. It should be noted, as shown in Fig. 7, that although sin(nπx)n=1 fails to converge at any point other than 0 and n−1 ∞ 1, the subsequence sin(2 πx)n=1 does converge graphically on a set dense in [0, 1].

2 1 1 3 1 5 functions. It is to be noted that the number of equiv- = c, c, c, c, c, c, alent fixed points in a class increases with the num- 3 4 8 ∓ 4 ∓ 8 ∓ 2 ∓ 8 ∓ ber of iterations k as 2k−1 + 1; this increase in the 3 7 c, c, 1 c degree of ill-posedness is typical of discrete chaotic 4 ∓ 8 ∓ − systems and can be regarded as a paradigm of chaos where c = 1/24. If the moduli of the slopes of the generated by the convergence of a family of func- graphs passing through these equivalent fixed points tions. are greater than 1 then the graphs converge to The mth iterate tm of the tent map has 2m fixed multifunctions and when these slopes are less than points corresponding to the 2m injective branches 1 the corresponding graphs converge to constant of tm

j 1 − , j = 1, 3, . . . , (2m 1) 2m 1 − x = tm(x ) = x , j = 1, 2, . . . , 2m . mj  j− mj mj  , j = 2, 4, . . . , 2m  2m + 1 m n Let Xm be the collection of these 2 fixed points higher iterates t for m = in with i = 1, 2, . . . (thus X1 = 0, 2/3 ), and denote by [Xm] the set where these subsequences remain fixed. For exam- of the equivalen{ t poin} ts, one coming from each of ple, the fixed points 2/5 and 4/5 produced respec- the injective branches, for each of the fixed points: tively by the second and fourth injective branches thus of t2, are also fixed for the seventh and thirteenth 4 1 1 branches of t . For the shift map 2xmod(1) on [0, 1], 2 ∞ m − = [X1] = [0], − = [0], [1] where [0] = m=1 (i 1)/2 : D 3 D { }m ∞ { −m i = 1, 2, . . . , 2 and [1] = m=1 i/2 : i = m } T { 2 2 4 1, 2, . . . , 2 . [X ] = [0], , , } T 2 5 3 5 It is useful to compare the graphical conver- ∞ gence of (sin(πnx))n=1 to [0, 1] at 0 and to 0 at 1 ∞ and + = m=1[Xm] is a non-empty countable with the usual integral and test-functional conver- set denseD in X at each of which the graphs of the gences to 0; note that the point 1/2, for example, sequence (tmT) converge to a multifunction. New belongs to and not to = 0, 1 because it D+ D− { } sets [Xn] will be formed by subsequences of the is frequented by even n only. However for the sub- December 3, 2003 12:13 00851

Toward a Theory of Chaos 3177

sequence (f m−1 ) Z , 1/2 is in because if the properties (ER1)–(ER3) of an equivalence relation, 2 m∈ + D− graph of f2m−1 passes through (1/2, 0) for some m, Tutorial 1) then so do the graphs for all higher values. There- ∞ m−1 m−1 fore [0] = m=1 i/2 : i = 0, 1, . . . , 2 is the (OR1) Reflexive, that is ( x X)(x x). { ∞ } ∀ ∈ equivalence class of (f2m−1 )m=1 and this sequence (OR2) Antisymmetric: ( x, y X)(x y y T converges to [ 1, 1] on this set. Thus our extension x x = y). ∀ ∈ ∧ Multi(X) is distinct− from the distributional exten- ⇒ (OR3) Transitive, that is ( x, y, z X)(x y y sion of function spaces with respect to test func- z x z). Any notion of∀order on∈ a setX in∧ the tions, and is able to correctly generate the patho- sense⇒ ofone element of X preceding another should logical behavior of the limits that are so crucially possess at least this property. vital in producing chaos. The relation is a preorder - if it is only reflexive 4. Discrete Chaotic Systems are and transitive, that is if only (OR1) and (OR3) are Maximally Ill-posed true. If the hypothesis of (OR2) is also satisfied by a preorder, then this - induces an equivalence re- The above ideas apply to the development of a cri- lation on X according to (x - y) (y - x) terion for chaos in discrete dynamical systems that x y ∼that evidently is actually a partial∧ order ⇔iff is based on the limiting behavior of the graphs of x ∼ y x = y. For any element [x] X/ of the ∼ ⇔ ∈ ∼ a sequence of functions (fn) on X, rather than on induced quotient space, let denote the generated the values that the sequence generates as is cus- order in X/ so that ≤ tomary. For the development of the maximality of ∼ ill-posedness criterion of chaos, we need to refresh x - y [x] [y] ; ⇔ ≤ ourselves with the following preliminaries. then is a partial order on X/ . If every two elements≤of X are comparable, in the∼ sense that either x1 x2 or x2 x1 for all x1, x2 X, then X is saidto be a total ly ordered set or∈a chain. A to- Resume Tutorial 5: Axiom of Choice tally ordered subset (C, ) of a partially ordered and Zorn’s Lemma set (X, ) with the ordering induced from X, is Let us recall from the first part of this Tutorial that known as a chain in X if for nonempty subsets (Aα)α∈D of a nonempty set C = x X : ( c X)(c x x c) . (37) X, the Axiom of Choice ensures the existence of a { ∈ ∀ ∈ ∨ } set A such that A Aα consists of a single element The most important class of chains that we are con- for every α. The choice axiom has far reaching concerned with in this work is that on the subsets (X) sequences and a fewT equivalent statements, one of of a set (X, ) under the inclusion order; Eq.P(37), which the Zorn’s lemma that will be used immedi- as we shall see⊆ in what follows, defines a family of ately in the following, is the topic of this resumed chains of nested subsets in (X). Thus while the Tutorial. The beauty of the Axiom, and of its equiv- relation - in Z defined by nP - n n n 1 2 ⇔ | 1| ≤ | 2| alents, is that they assert the existence of mathe- with n1, n2 Z preorders Z, it is not a partial matical objects that, in general, cannot be demon- order because∈ although n - n and n - n for strated and it is often believed that Zorn’s lemma any n Z, it is does not− follow that n−= n. is one of the most powerful tools that a mathemati- A common∈ example of partial order on−a set of cian has available to him that is “almost indispens- sets, for example on the power set (X) of a set able in many parts of modern pure mathematics” X (see footnote 23), is the inclusion relationP : the with significant applications in nearly all branches ordered set = ( ( x, y, z ), ) is partially⊆ or- of contemporary mathematics. This “lemma” talks dered but notX totallyP { ordered} b⊆ecause, for exam- about maximal (as distinct from “maximum”) ele- ple, x, y y, x , or x is not comparable to ments of a partially ordered set, a set in which some y unless{ } x6⊆={ y; ho} wev{er }C = , x , x, y { } {{∅ { } { }} notion of x1 “preceding” x2 for two elements of the does represent one of the many possible chains of set has been defined. . Another useful example of partial order is the A relation on a set X is said to be a partial Xfollowing: Let X and (Y, ) be sets with or- order (or simplyan order) if it is (compare with the dering Y , and consider f,≤g Map(X, Y )≤with ∈ December 3, 2003 12:13 00851

3178 A. Sengupta

(f), (g) X. Then which requires the upper bound u to be larger than D D ⊆ all members of A, with the corresponding lower ( (f) (g))(f = g ) f g D ⊆ D |D(f) ⇔ bounds of A being defined in a similar manner. Of ( (f) = (g))( (f) (g)) f g (38) course, it is again not necessary that the elements D D R ⊆ R ⇔ ( x (f) = (g))(f(x) g(x)) f g of A be comparable to each other, and it should ∀ ∈ D D ≤ ⇔ be clear from Eqs. (41) and (42) that when an up- define partial orders on Map(X, Y ). In the last case, per bound of a set is in the set itself, then it is the the order is not total because any two functions maximum element of the set. If the upper (lower) whose graphs cross at some point in their common bounds of a subset (A, ) of a set (X, ) has a least domain cannot be ordered by the given relation, (greatest) element, then this smallestupper bound while in the first any f whose graph does not coin- (largest lower bound) is called the least upper bound cide with that of g on the common domain is not (greatest lower bound) or supremum (infimum) of A comparable to it by this relation. in X. Combining Eqs. (41) and (42) then yields Let (X, ) be a partially ordered set and let A sup A = a← ΩA : a← u u (ΩA, ) be a subset of X. An element a+ (A, ) is said X { ∈ ∀ ∈ } to be a maximal element of A with∈respect to if (43) inf A = →a ΛA : l → a l (ΛA, ) X ( a (A, ))(a+ a) a = a+ , (39) { ∈ ∀ ∈ } ∀ ∈ ⇒ where Ω = u X : ( a A)(a u) and that is, iff there is no a A with a = a and A + Λ = l X :{( a∈ A)(l ∀ a)∈ are thesets}of all a a .21 Expressed otherwise,∈ this implies6 that an A + upper {and∈ lower∀bounds∈ ofA in} X. Equation (43) elemen t a of a subset A (X, ) is maximal in + may be expressed in the equivalent but more trans- (A, ) iff it is true that ⊆ parent form as (a a+ A)(for every a (A, ) a = sup A (a A a a ) ∈ ∈ ← ⇔ ∈ ⇒ ← comparable to a+) ; (40) X (a0 a← a0 b a← for some b A) thus a+ in A is a maximal element of A iff it is ∧ ≺ ⇒ ≺ ∈ →a = inf A (a A → a a) strictly greater than every other comparable element X ⇔ ∈ ⇒ of A. This of course does not mean that each ele- (→a a1 → a b a1 for some b A) ment a of A satisfies a a because every pair ∧ ≺ ⇒ ≺ ∈ + (44) of elements of a partiallyordered set need not be comparable: in a totally ordered set there can be to imply that a← (→a) is the upper (lower) bound of at most one maximal element. In comparison, an A in X which precedes (succeeds) every other upper element a∞ of a subset A (X, ) is the unique (lower) bound of A in X. Notice that uniqueness in maximum (largest, greatest,⊆last) elemen t of A iff the definitions above is a direct consequence of the uniqueness of greatest and least elements of a set. (a a A)(for every a (A, )) , (41) ∞ ∈ ∈ It must be noted that whereas maximal and max- implying that a∞ is the element of A that is strictly imum are properties of the particular subset and larger than every other element of A. As in the case have nothing to do with anything outside it, up- of the maximal, although this also does not require per and lower bounds of a set are defined only with all elements of A to be comparable to each other, respect to a superset that may contain it. it does require a∞ to be larger than every element The following example, beside being useful in of A. The dual concepts of minimal and minimum Zorn’s lemma, is also of great significance in fix- can be similarly defined by essentially reversing the ing some of the basic ideas needed in our future roles of a and b in relational expressions like a b. arguments involving classes of sets ordered by the The last concept needed to formalize Zorn’s inclusion relation. lemma is that of an upper bound: For a subset Example 4.1. Let = ( a, b, c ) be ordered (A, ) of a partially ordered set (X, ), an element by the inclusion relationX P {. The }subset = u ofX is an upper bound of A in X iff ( a, b, c ) a, b, c has three⊆ maximals Aa, b , (a u (X, ))(for every a (A, )) (42) Pb,{c and} −c, {a but}no maximum as there{ is no} ∈ ∈ { } { } 21If is an order relation in X then the strict relation in X corresponding to , given by x y (x y) (x = y), is ≺ ≺ ⇔ ∧ 6 not an order relation because unlike , is not reflexive even though it is both transitive and asymmetric. ≺ December 3, 2003 12:13 00851

Toward a Theory of Chaos 3179

A∞ satisfying A A∞ for every A , The statement of Zorn’s lemma and its proof while∈ A( a, b, c ) the three minimals a ∈, Ab can now be completed in three stages as follows. and cP {but no}minim− ∅ um. This shows that{ a} sub-{ } For Theorem 4.1 below that constitutes the most set of{ a} partially ordered set may have many max- significant technical first stage, let g be a function imals (minimals) without possessing a maximum on (X, ) that assigns to every x X an immediate (minimum), but a subset has a maximum (mini- successor y X such that ∈ mum) iff this is its unique maximal (minimal). If ∈ (x) = y x : x X satisfying x x y = a, b , a, c , then every subset of the in- M { 6 ∃ ∗ ∈ ≺ ∗ ≺ } A {{ } { }} tersection of the elements of , namely a and , are all the successors of x in X with no element A { } ∅ are lower bounds of , and all supersets in of the of X lying strictly between x and y. Select a rep- A X union of its elements — which in this case is just resentative of M(x) by a choice function fC such a, b, c — are its upper bounds. Notice that while that the{ maximal} (minimal) and maximum (minimum) g(x) = f ( (x)) M(x) are elements of , upper and lower bounds need C M ∈ A not be contained in their sets. In this class ( , ) is an immediate successor of x chosen from the X ⊆ of subsets of a set X, X+ is a maximal element of many possible in the set M(x). The basic idea in iff X+ is not contained in any other subset of X, the proof of the first of the three-parts is to express X while X∞ is a maximum of iff X∞ contains every the existence of a maximal element of a partially X other subset of X. ordered set X in terms of the existence of a fixed Let := Aα α∈D be a non-empty sub- point in the set, which follows as a contradiction A { ∈ X } class of ( , ), and suppose that both Aα and of the assumed hypothesis that every point in X X ⊆ Aα are elements of . Since each Aα is -less has an immediate successor. Our basic application X S ⊆ than Aα, it follows that Aα is an upper bound of immediate successors in the following will be to T of ; this is also the smallest of all such bounds classes ( (X), ) of subsets of a set X or- A S S because if U is any other upper bound then every dered byXinclusion.⊆ P In⊆this case for any A , the ∈ X Aα must precede U by Eq. (42) and therefore so function g can be taken to be the superset must Aα (because the union of a class of subsets g(A) = A fC( (A) A) , of a set is the smallest that contain each member G − (46) of theSclass: A U A U for subsets where (A) = x X A : A x α α G { S∈ − { } ∈ X } (A ) and U of X⊆). Analogously⇒ , since⊆ A is - α α of A. Repeated application of g to A then generates less than each A it is a lowSer bound of ; that⊆it S α a principal filter, and hence an associated sequence, is the greatest of all the lower bounds LTAin fol- based at A. lows because the intersection of a class of subsetsX is the largest that is contained in each of the subsets: Theorem 4.1. Let (X, ) be a partially ordered set L Aα L Aα for subsets L and (Aα) of X. that satisfies Hence⊆ the⇒ suprem⊆ um and infimum of in ( , ) (ST1) There is a smallest element x of X which given by T A X ⊆ 0 has no immediate predecessor in X. (ST2) If C X is a totally ordered subset in X, A← = sup = A (X ,⊆) A then c = sup⊆ C is in X. A∈A ∗ X [ (45) Then there exists a maximal element x+ of X which and →A = inf = A has no immediate successor in X. (X ,⊆) A A\∈A Proof. Let T (X, ) be a subset of X. If the con- are both elements of ( , ). Intuitively, an upper ⊆ (respectively, lower) boundX ⊆of in is any subset clusion of the theorem is false then the alternative A X of that contains (respectively, is contained in) (ST3) Every element x T has an immediate suc- X every member of . cessor g(x) in T 22 ∈ A

22This makes T , and hence X, inductively defined infinite sets. It should be realized that (ST3) does not mean that every member of T is obtained from g, but only ensures that the immediate successor of any element of T is also in T. The infimum →T of these towers satisfies the additional property of being totally ordered (and is therefore essentially a sequence or net) in (X, ) to which (ST2) can be applied. December 3, 2003 12:13 00851

3180 A. Sengupta

leads, as shown below, to a contradiction that can g(c) t then g(c) g(t); this combined with be resolved only by the conclusion of the theo- (c = t) (g(c) = g(t≺)) yields g(c) g(t). On the rem. A subset T of (X, ) satisfying conditions other hand,⇒ t c for every t C requires g(t) c g (ST1) (ST3) is sometimes known as an g-tower as otherwise (≺t c) (c ∈g(t)) would, from the − or an g-sequence: an obvious example of a tower resulting consequence≺ ⇒t ≺c g(t), contradict the is (X, ) itself. If assumed hypothesis that≺g(t)≺is the immediate successor of t. Hence, Cg is a g-tower in X. →T = T : T is an x0 tower To complete the proof that g(c) C , and { ∈ T − } ∈ T is the ( (X),\ )-infimum of the class of all se- thereby the argument that CT is a tower, we first quentialPtowers⊆of (X, ), we show thatTthis small- note that as →T is the smallest tower and Cg is built est sequential tower isinfact a sequential totally from it, Cg =→ T must infact be →T itself. From Eq. (48) therefore, for every t → T either t g(c) ordered chain in (X, ) built from x0 by the g- or g(c) t, so that g(c) C∈ whenever c C . function. Let the subset ∈ T ∈ T This concludes the proof that CT is actually the CT = c X : ( t → T )(t c c t) X (47) tower T in X. { ∈ ∀ ∈ ∨ } ⊆ → From (ST2), the implication of the chain CT of X be an g-chain in →T in the sense that [cf. Eq. (37)] it is that subset of X each of whose CT =→ T = Cg (49) elements is comparable with some element of →T . being the minimal tower →T is that the supre- The conditions (ST1)–(ST3) for CT can be verified mum t← of the totally ordered →T in its own as follows to demonstrate that CT is an g-tower. tower (as distinct from in the tower X: recall that

(1) x0 CT, because it is less than each x →T . →T is a subset of X) must be contained in itself, ∈ ∈ that is (2) Let c← = supX CT be the supremum of the chain C in X so that by (ST2), c X. Let T ← ∈ sup(CT) = t← → T X . (50) t T . If there is some c C such that t c, CT ∈ ⊆ ∈ → ∈ T then surely t c←. Else, c t for every c CT This however leads to the contradiction from shows that c t because c is the small-∈ ← ← (ST3) that g(t←) be an element of →T , unless of est of all the upper bounds t of CT. Therefore course c← CT. (3) In order∈ to show that g(c) C whenever c C g(t←) = t← , (51) ∈ ∈ it needs to be verified that for all t T , ei- which because of (49) may also be expressed equiv- ∈ → ther t c t g(c) or c t g(c) t. alently as g(c ) = c C . As the sequential ⇒ ⇒ ← ← ∈ T As the former is clearly obvious, we investigate totally ordered set →T is a subset of X, Eq. (48) the latter as follows; note that g(t) T by implies that t is a maximal element of X which ∈ → ← (ST3). The first step is to show that the subset allows (ST3) to be replaced by the remarkable inverse criterion that C = t T : ( c C )(t c g(c) t) g { ∈ → ∀ ∈ T ∨ } (ST30) If x X and w precedes x, w x, (48) then w X, that∈ is obviously false for a general≺ ∈ of T , which is a chain in X (observe the in- tower T . In fact, it follows directly from Eq. (39) → 0 verse roles of t and c here as compared to that in that under (ST3 ) any x+ X is a maximal element of X iff it is a fixed ∈point of g as given by Eq. (47)), is a tower: Let t← be the supremum Eq. (51). This proves the theorem and also demon- of Cg and take c C. If there is some t Cg ∈ ∈ strates how, starting from a minimum element of a for which g(c) t, then clearly g(c) t←. Else, partially ordered set X, (ST3) can be used to gen- t x for each t Cg shows that t← c be- ∈ erate inductively a totally ordered sequential subset cause t← is the smallest of all the upper bounds of X leading to a maximal x = c (X, ) that c of Cg. Hence t← Cg. + ← ∈ is a fixed point of the generating function∈ gwhen- Property (ST3) for Cg follows from a small ever the supremum t← of the chain →T is in X. yet significant modification of the above arguments in which the immediate successors g(t) of t C ∈ g formally replaces the supremum t← of Cg. Thus Remark. The proof of this theorem, despite its ap- given a c C, if there is some t C for which parent length and technically involved character, ∈ ∈ g December 3, 2003 12:13 00851

Toward a Theory of Chaos 3181

carries the highly significant underlying message be the set of all the totally ordered subsets of that (X, ). Since is a collection of (sub)sets of X, we order it byXthe inclusion relation on and use Any inductive sequential g-construction of the tower Theorem to demonstrate that (X , ) has an infinite chained tower C starting with X ⊆ T a maximal element C←, which by the definition of a smallest element x0 (X, ) such that , is the required maximal chain in (X, ). ∈ X a supremum c← of the g-generated sequen- Let be a chain in of the chains in (X, ). C X tial chain CT in its own tower is contained In order to apply the tower Theorem to ( , ) we in itself, must necessarily terminate with a need to verify hypothesis (ST2) that the smallestX ⊆ fixed point relation of the type (51) with respect to the supremum. Note from Eqs. (50) C∗ = sup = C (53) X C and (51) that the role of (ST2) applied to C[∈C a fully ordered tower is the identification of of the possible upper bounds of [see Eq. (45)] is the maximal of the tower — which depends C a chain of (X, ). Indeed, if x1, x2 X are two only on the tower and has nothing to do ∈ points of Csup with x1 C1 and x2 C2, then with anything outside it — with its supre- ∈ ∈ from the -comparability of C1 and C2 we may mum that depends both on the tower and its ⊆ choose x1, x2 C1 C2, say. Thus x1 and x2 are complement. ∈ ⊇ -comparable as C1 is a chain in (X, ); C∗ is therefore a chain in (X, ) which establishes ∈thatX Thus although purely set-theoretic in nature, the the supremum of a chainof ( , ) is a chain in filter-base associated with a sequentially totally or- (X, ). X ⊆ dered set may be interpreted to lead to the usual The tower Theorem 4.1 can now be applied to notions of adherence and convergence of filters and ( , ) with C as its smallest element to construct thereby of a generated topology for (X, ), see 0 aXg-sequen⊆ tially towered fully ordered subset of Appendix A.1 and Example A.1.3. This very sig- consisting of chains in X X nificant apparent inter-relation between topologies, filters and orderings will form the basis of our = C (X) : C C for i j N CT { i ∈ P i ⊆ j ≤ ∈ } approach to the condition of maximal ill-posedness = (X) for chaos. →T ⊆ P In the second stage of the three-stage of ( , ) — consisting of the common elements of X ⊆ programme leading to Zorn’s lemma, the tower all g-sequential towers T of ( , ) — that in- T ∈ X ⊆ Theorem 4.1 and the comments of the preceding fact is a principal filter base of chained subsets of paragraph are applied at a higher level to a very (X, ) at C0. The supremum (chain in X) C← of T C special class of the power set of a set, the class of in T must now satisfy, by Theorem 4.1, the fixed C all the chains of a partially ordered set, to directly point g-chain of X lead to the physically significant sup( T) = C← = g(C←) T (X) , CT C ∈ C ⊆ P Theorem 4.2 (Hausdorff Maximal Principle). where the chain g(C) = C f ( (C) C) with Every partially ordered set (X, ) has a maximal C (C) = x X C : C x G ,−is an im- totally ordered subset.23 Gmediate successor{ ∈ −of C obtainedS{ } ∈byXc}hoosing one S Proof. Here the base level is point x = fC( (C) C) from the many possible in (C) C sucGh that−the resulting g(C) = C x = C (X) : C is a chain in (X, ) (X) G − { } X { ∈ P } ⊆ P is a strict successor of the chain C with no others (52) lying between it and C. Note that C ( , S ) is ← ∈ X ⊆ 23 Recall that this means that if there is a totally ordered chain C in (X, ) that succeeds C+, then C must be C+ so that no chain in X can be strictly larger than C+. The notation adopted here and below is the following: If X = x, y is a non-empty { } set, then := (X) = A : A X = , x , y , x, y is the set of subsets of X, and X := 2(X) = : , the set X P { ⊆ } {∅ { } { } { }} P {A A ⊆ X } of all subsets of , consists of the 16 elements , , x , y , x, y , , x , , y , , x, y , x , y , X ∅ {∅} {{ }} {{ }} {{ }} {{∅} { }} {{∅} { }} {{∅} { }} {{ } { }} x , x, y , y , x, y , , x , y , , x , x, y , , y , x, y , x , y , x, y , and : an element {{ } { }} {{ } { }} {{∅} { } { }} {{∅} { } { }} {{∅} { } { }} {{ } { } { }} X of 2(X) is a subset of (X), any element of which is a subset of X. Thus if C = 0, 1, 2 is a chain in (X = 0, 1, 2 , ), P P { } { } ≤ then = 0 , 0, 1 , 0, 1, 2 (X) and C = 0 , 0 , 0, 1 , 0 , 0, 1 , 0, 1, 2 2(X) represent chains C {{ } { } { }} ⊆ P {{{ }} {{ } { }} {{ } { } { }}} ⊆ P in ( (X), ) and ( 2(X), ), respectively. P ⊆ P ⊆ December 3, 2003 12:13 00851

3182 A. Sengupta

(X, ) X = {C ⊆ X : C is a chain in (X, )}

Tower Theorem 4.1

CT = {T ⊆ (X , ⊆): T is a C0 tower}

supC (C )=C←=g(C←)∈C ⊆(X,⊆) T T T Hausdorﬀ Maximal Chain Theorem Zorn Lemma

(u ∈ X c)(∀c∈(C←,))

Fig. 10. Application of Zorn’s Lemma to (X, ). Starting with a partially ordered set (X, ), construct: (a) The one-level higher subset = C (X) : C is a chain in (X, ) of (X) consisting of all the totally ordered subsets X { ∈ P } P of (X, ), (b) The smallest common g-sequential totally ordered towered chain = Ci (X) : Ci Cj for i j (X) of all CT { ∈ P ⊆ ≤ } ⊆ P sequential g-towers of by Theorem 4.1, which in fact is a principal filter base of totally ordered subsets of (X, ) at the X smallest element C0. (c) Apply Hausdorff Maximal Principle to ( , ) to get the subset sup ( ) = C← = g(C←) (X) of (X, ) as the X ⊆ CT CT ∈ CT ⊆ P supremum of ( , ) in . The identification of this supremum as a maximal element of ( , ) is a consequence of (ST2) X ⊆ CT X ⊆ and Eqs. (50), (51) that actually puts the supremum into itself. X By returning to the original level (X, ) (d) Zorn’s Lemma finally yields the required maximal element u X as an upper bound of the maximal totally ordered subset ∈ (C←, ) of (X, ). The dashed segment denotes the higher Hausdorff ( , ) level leading to the base (X, ) Zorn level. X ⊆

only one of the many maximal fully ordered subsets Indeed, if there is an element v X that is compa- ∈ possible in (X, ). rable to u and v u, then v cannot be in C← as it is necessary for every6 x C to satisfy x u. Clearly With the assurance of the existence of a max- ← then C v is a chain∈ in (X, ) bigger than C imal chain C among all fully ordered subsets of ← ← ← which contradicts{ } the assumed maximality of C a partially ordered set (X, ), the arguments are ← among theS chains of X. completed by returning to the basic level of X. 1 The sequence of steps leading to Zorn’s Lemma, Theorem 4.3 (Zorn’s Lemma). Let (X, ) be a and thence to the maximal of a partially ordered set, partially ordered set such that every totallyordered is summarized in Fig. 10. subset of X has an upper bound in X. Then X The three examples below of the application of has at least one maximal element with respect to Zorn’s Lemma clearly reflect the increasing com- its order. plexity of the problem considered, with the maxi- Proof. The proof of this final part is a mere ap- mals a point, a subset, and a set of subsets of X, plication of the Hausdorff Maximal Principle on so that these are elements of X, (X) and 2(X), P P the existence of a maximal chain C← in X to the respectively. hypothesis of this theorem that C has an upper ← Example 4.2 bound u in X that quickly leads to the identification of this bound as a maximal element x of X. (1) Let X = ( a, b, c , ) be a three-point base- + { } December 3, 2003 12:13 00851

Toward a Theory of Chaos 3183

¢£ ¢¡

§¦ ¨ ¢¤ ©

¤ ¡

(a) (b)

Fig. 11. Tree diagrams of two partially ordered sets where two points are connected by a line iﬀ they are comparable to each other, with the solid lines linking immediate neighbors and the dashed, dotted and dashed–dotted lines denoting second, third and fourth generation orderings according to the principle of transitivity of the order relation. There are 8 2 chains of × (a) and 7 chains of (b) starting from respective smallest elements with the immediate successor chains shown in solid lines. The 17 point set X = 0, 1, 2, . . . , 15, 16 in (a) has two maximals but no maximum, while in (b) there is a single maximum { } of ( a, b, c ), and three maximals without any maximum for ( a, b, c ) a, b, c . In (a), let A = 1, 3, 4, 7, 9, 10, 15 , P { } P { } − { } { } B = 1, 3, 4, 6, 7, 13, 15 , C = 1, 3, 4, 10, 11, 16 and D = 1, 3, 4 . The upper bounds of D in A are 7, 10 and 15 without { } { } { } any supremum (as there is no smallest element of 7, 10, 15 ), and the upper bounds of D in B are 7 and 15 with sup (D) = 7, { } B while sup (D) = 10. Finally the maximal, maximum and the supremum in A of 1, 3, 4, 7 are all the same illustrating how C { } the supremum of a set can belong to itself. Observe how the supremum and upper bound of a set are with reference to its complement in contrast with the maximum and maximal that have nothing to do with anything outside the set.

level ground set ordered lexicographically, that is of , with a b c. A chain of the partially ordered X ≺ ≺ C sup( T) = C← = a, b, c = g(C←) T (X) Hausdorff-level set consisting of subsets of X CT C { } ∈ C ⊆ P given by Eq. (52) is,Xfor example, a , a, b and the six g-sequential chained towers{{ } { }} the only maximal element of (X). Zorn’s Lemma now assures the existence of aPmaximal element of 1 = , a , a, b , a, b, c , c X. Observe how the maximal element of (X, ) C {∅ { } { } { }} is∈obtained by going one level higher to at the 2 = , a , a, c , a, b, c C {∅ { } { } { }} Hausdorff stage and returning to the baseXlevel X = , b , a, b , a, b, c , C3 {∅ { } { } { }} at Zorn, see Fig. 10 for a schematic summary of this 4 = , b , b, c , a, b, c sequence of steps. C {∅ { } { } { }} 5 = , c , a, c , a, b, c , (2) Basis of a vector space. A linearly independent C {∅ { } { } { }} 6 = , c , b, c , a, b, c set of vectors in a vector space X that spans the C {∅ { } { } { }} space is known as the Hamel basis of X. To prove built from the smallest element corresponding to the existence of a Hamel basis in a vector space, ∅ the six distinct ways of reaching a, b, c from Zorn’s lemma is invoked as follows. { } ∅ along the sides of the cube marked on the figure The ground base level of the linearly indepen- with solid lines, all belong to ; see Fig. 11(b). dent subsets of X An example of a tower in ( X, ) which is not X ⊆ J J a chain is = , a , b , c , a, b , a, c , = xij j=1 (X) : Span( xij j=1) T {∅ { } { } { } { } { } X {{ } ∈ P { } b, c , a, b, c . Hence the common infimum tow- J = 0 (αj)j=1 = 0 J 1 (X) , {ered }chained{ subset}} ⇒ ∀ ≥ } ⊆ P with Span( x J ) := J α x , is such that no { ij }j=1 j=1 j ij T = , a, b, c =→ (X) x can be expressed as a linear combination of C {∅ { }} T ⊆ P ∈ X P December 3, 2003 12:13 00851

3184 A. Sengupta

the elements of x . clearly has a smallest Compared to this purely algebraic concept of X − { } X element, say xi1 , for some non-zero xi1 X. Let basis in a vector space, is the Schauder basis in the higher Hausdorff{ } level ∈ a normed space which combines topological structure with the linear in the form of convergence: If X = 2(X) : is a chain in ( , ) 2(X) {C ∈ P C X ⊆ } ⊆ P a normed vector space contains a sequence (ei)i∈Z+ and collection of the chains with the property that for every x X there is an ∈ unique sequence of scalars (αi)i∈Z+ such that the iK = xi1 , xi1 , xi2 , . . . , xi1 , xi2 , . . . , xiK C {{2 } { } { }} remainder x (α1e1 + α2e2 + + αI eI ) ap- (X) proaches 0kas I− , then the·collection· · (ek ) is ∈ P → ∞ i of comprising linearly independent subsets of X known as a Schauder basis for X. X C be g-built from the smallest xi1 . Any chain of (3) Ultrafilter. Let X be a set. The set = { } FS X is bounded above by the union ∗ = supX C = S (X) : S S = , α = β (X) of all C { α ∈ P α β 6 ∅ ∀ 6 } ⊆ P C which is a chain in containing xi1 , nonempty subsets of X with finite intersection prop- C∈ C X { } thereby verifying (ST2) for X. Application of the erty is known asTa filter subbase on X and = S F tower theorem to X implies that the element B X : B = S , for I D a finite subsetB { ⊆ i∈I⊂D i} ⊂ C T 2 of a directed set D, is a filter-base on X associated T = i1 , i2 , . . . , in , . . . =→ (X) T {C C C } ⊆ P with the subbase F ; cf. Appendix A.1. Then the X S in of chains of is a g-sequential fully ordered filter generated by F consisting of every superset towered subset ofX(X, ) consisting of the com- S ⊆ of the finite intersections B F of sets of F is mon elements of all g-sequential towers of (X, ), the smallest filter that contains∈ theB subbase Sand ⊆ F that in fact is a chained principal ultrafilter on base . For notational simplicity, we will denoteS FB ( (X), ) generated by the filter-base xi1 at the subbase in the rest of this example simply P ⊆ {{{ }}} FS xi1 , where by . { } S T Consider the base-level ground set of all filter = i1 , i2 , . . . , jn , jn+1 , . . . {C C C C } subbases on X for some n N is an example of non-chained g- tower whenev∈er ( )∞ is neither contained in nor Cjk k=n S = 2(X) : = for every finite subset of contains any member of the ( )∞ chain. Haus- S ∈P R 6 ∅ S ik k=1 ( ∅6=R⊆S ) dorff’s chain theorem now yieldsC the fixed-point g- \ 2(X), chain X of ⊆ P C← ∈ X C ordered by inclusion in the sense that α β for sup( T)= ← = xi1 , xi1 , xi2 , xi1 , xi2 , xi3 , . . . S ⊆ S C C {{ } { } { } } all α β D, and let the higher Hausdorff-level T ∈ C 2 =g( ←) T (X) X˜ = C 3(X) : C is a chain in (S, ) 3(X) C ∈ ⊆P { ∈ P ⊆ } ⊆ P as a maximal totally ordered principal filter on X comprising the collection of the totally ordered

that is generated by the filter-base xi1 at xi1 , chains whose supremum B = x , x , . . . {{ (}}X) is, by { i1 i2 } ∈ P C = S , S , S , . . . , S , S , . . . , S Zorn’s lemma, a maximal element of the base level κ {{ α} { α β} { α β κ}} . This maximal linearly independent subset of X 3(X) ∈ P Xis the required Hamel basis for X: Indeed, if the of S be g-built from the smallest S then an ultra- span of B is not the whole of X, then Span(B) x, α filter on X is a maximal member { of} (S, ) in the with x / Span(B) would, by definition, be a linearly + usual sense that any subbase onS X must necessar-⊆ independen∈ t set of X strictly larger than B, con-S ily be contained in so thatS = tradicting the assumed maximality of the later. It + + + for any (X) Swith FIP. TheS ⊆toSwer⇒ theoremS S needs to be understood that since the infinite basis now impliesS ⊆thatP the element cannot be classified as being linearly independent, 3 we have here an important example of the supre- C˜T = Cα, Cβ, . . . , Cν, . . . = →˜T (X) mum of the maximal chained set not belonging to { } ⊆ P the set even though this criterion was explicitly used of 4(X), which is a chain in X˜ of the chains of S, in the construction process according to (ST2) and is aPg-sequential fully ordered towered subset of the (ST3). common elements of all sequential towers of (X˜, ) ⊆ December 3, 2003 12:13 00851

Toward a Theory of Chaos 3185

that is a chained principal ultrafilter on ( 2(X), ) element of (X, ). This sequence is now applied, as P ⊆ generated by the filter-base Sα at Sα , where in Example 4.2(1), to the set of arbitrary relations {{{ }}} { } Multi(X) on an infinite set X in order to formulate T˜ = C , C , . . . , C , C , . . . , { α β σ ς } our definition of chaos that follows. is an obvious example of non-chained g-tower when- Let f be a noninjective map in Multi(X) and ever (Cσ) is neither contained in, nor contains, any P (f) the number of injective branches of f. Denote member of the Cα-chain. Hausdorff’s chain theorem by now yields the fixed-point C˜← X˜ ∈ F = f Multi(X) : f is a noninjective function sup(C˜T) = C˜← = Sα , Sα, Sβ , Sα, Sβ, Sγ , . . . { ∈ ˜ {{ } { } { } } on X Multi(X) CT } ⊆ = g(C˜ ) C˜ 3(X) ← ∈ T ⊆ P the resulting basic collection of noninjective func- as a maximal totally ordered g-chained towered sub- tions in Multi(X). set of X that is, by Zorn’s lemma, a maximal ele- 2 (i) For every α in some directed set D, let F have ment of the base level subset S of (X). C˜← is a chained principal ultrafilter on ( (PX), ) gener- the extension property P ⊆ ated by the filter-base Sα at Sα, while + = ( fα F )( fβ F ) : P (fα) P (fβ) 2 {{ }} S Sα, Sβ, Sγ, . . . (X) is an (non-principal) ul- ∀ ∈ ∃ ∈ ≤ tr{ afilter on X —} c∈haracterizedP by the property that (ii) Let a partial order on Multi(X) be defined, for f , f Map(X) Multi(X) by any collection of subsets on X with FIP (that is any α β ∈ ⊆ filter subbase on X) must be contained in the max- P (f ) P (f ) f f , (54) imal set having FIP — that is not a principal α ≤ β ⇔ α β S+ filter unless α is a singleton set xα . with P (f) := 1 for the smallest f, define a par- S { } tially ordered subset (F, ) of Multi(X). This What emerges from these applications of Zorn’s is actually a preorder onMulti(X) in which Lemma is the remarkable fact that infinities (the functions with the same number of injective dot-dot-dots) can be formally introduced as “limit- branches are equivalent to each other. ing cases” of finite systems in a purely set-theoretic (iii) Let context without the need for topologies, metrics or C = f Multi(X) : f f (F ) , convergences. The significance of this observation ν { α ∈ α ν} ∈ P will become clear from our discussions on filters and ν D , ∈ topology leading to Sec. 4.2 below. Also, the obser- be g-chains of non-injective functions of vation on the successive iterates of the power sets Multi(X) and (X) in the examples above was to suggest their P anticipated role in the complex evolution of a dy- = C (F ) : C is a chain in (F, ) (F ) namical system that is expected to play a significant X { ∈ P }⊆P part in our future interpretation and understanding denote the corresponding Hausdorff level of all of this adaptive and self-organizing phenomenon of chains of F , with nature. = C , C , . . . , C , . . . = (F ) CT { α β ν } → T ⊆ P End Tutorial 5 being a g-sequential in . By Hausdorff Max- imal Principle, there is Xa maximal fixed-point g-towered chain C of F ← ∈ X From the examples in Tutorial 5, it should be clear sup( T) = C← = fα, fβ, . . . that the sequential steps summarized in Fig. 10 are CT C { } involved in an application of Zorn’s lemma to show = g(C ) (F ). that a partially ordered set has a maximal element ← ∈ CT ⊆ P with respect to its order. Thus for a partially or- Zorn’s Lemma applied to this maximal chain yields dered set (X, ), form the set of all chains C in its supremum as the maximal element of C , and X ← X. If C+ is a maximal chain of X obtained by the thereby of F . It needs to be appreciated, as in the Hausdorff Maximal Principle from the chain of case of the algebraic Hamel basis, that the exis- all chains of X, then its supremum u is a maximalC tence of this maximal non-functional element was December 3, 2003 12:13 00851

3186 A. Sengupta

obtained purely set theoretically as the “limit” of a [Devaney, 1989] and is also maximally non-injective; net of functions with increasing nonlinearity, with- the tent map is therefore chaotic on +. In con- out resorting to any topological arguments. Because trast, the examples of Secs. 1 and 2 areDnot chaotic it is not a function, this supremum does not be- as the maps are not topologically transitive, al- long to the functional g-towered chain having it though the Liapunov exponents, as in the case of as a fixed point, and this maximal chain does not the tent map, are positive. Here the (fn) are iden- possess a largest, or even a maximal, element, al- tified with the iterates of f, and the “fixed point” though it does have a supremum.24 The supremum as one through which graphs of all the functions on is a contribution of the inverse functional relations residual index subsets pass. When the set of points − (fα ) in the following sense. From Eq. (2), the net + is dense in [0, 1] and both + and [0, 1] + = D ∞ −i D − D of increasingly non-injective functions of Eq. (54) [0, 1] i=0 f (Per(f)) (where Per(f) denotes the implies a corresponding net of increasingly multi- set of−periodic points of f) are totally disconnected, valued functions ordered inversely by the inverse it is expSected that at any point on this complement relation f f f − f −. Thus the inverse re- the behavior of the limit will be similar to that on α β ⇔ β α lations which are as much an integral part of graph- +: these points are special as they tie up the iter- ical convergence as are the direct relations, have a atesD on Per(f) to yield the multifunctions. There- smallest element belonging to the multifunctional fore in any neighborhood U of a +-point, there D i class. Clearly, this smallest element as the required is an x0 at which the forward orbit f (x0) i≥0 is supremum of the increasingly non-injective tower chaotic in the sense that { } of functions defined by Eq. (54), serves to complete the significance of the tower by capping it with a (a) the sequence neither diverges nor does it con- “boundary” element that can be taken to bridge the verge in the image space of f to a periodic orbit classes of functional and non-functional relations of any period, and on X. (b) the Liapunov exponent is given by We are now ready to define a maximally ill- n 1/n posed problem f(x) = y for x, y X in terms of a def df (x0) ∈ λ(x0) = lim ln maximally non-injective map f as follows. n→∞ dx

n −1 Definition 4.1 (Chaotic map). Let A be a non- 1 df (xi) i = lim ln , xi = f (x0) , empty closed set of a compact Hausdorff space X. A n→∞ n dx i=0 function f Multi(X) equivalently the sequence of X ∈ functions (fi) is maximally non-injective or chaotic which is a measure of the a verage slope of an orbit on A with respect to the order relation (54) if at x0 or equivalently of the average loss of informa- tion of the position of a point after one iteration, is (a) for any f on A there exists an f on A satisfying i j positive. Thus an orbit with positive Liapunov expo- f f for every j > i N. i j nent is chaotic if it is not asymptotic (that is neither (b) theset consists of a∈countable collection of + convergent nor adherent, having no convergent sub- isolatedDsingletons. orbit in the sense of Appendix A.1) to an unstable Definition 4.2 (Maximally ill-posed problem). periodic orbit or to any other limit set on which the Let A be a non-empty closed set of a compact Haus- dynamics is simple. A basic example of a chaotic dorff space X and let f be a functional relation in orbit is that of an irrational in [0, 1] under the shift Multi(X). The problem f(x) = y is maximally ill- map and that of the chaotic set its closure, the full posed at y if f is chaotic on A. unit interval. Let f Map((X, )) and suppose that A = j ∈ U As an example of the application of these def- f (x0) j∈N is a sequential set corresponding to the { } j initions, on the dense set +, the tent map sat- orbit Orb(x0) = (f (x0))j∈N, and let fRi (x0) = D j isfies both the conditions of sensitive dependence j≥i f (x0) be the i-residual of the sequence j on initial conditions and topological transitivity (f (x )) N, with = fR (x ) : Res(N) X S 0 j∈ FBx0 { i 0 → 24 A similar situation arises in the following more intuitive example. Although the subset A = 1/n n Z of the interval { } ∈ + I = [ 1, 1] has no smallest or minimal elements, it does have the infimum 0. Likewise, although A is bounded below by any − element of [ 1, 0), it has no greatest lower bound in [ 1, 0) (0, 1]. − − S December 3, 2003 12:13 00851

Toward a Theory of Chaos 3187

for all i N being the decreasingly nested filter- It is important that the difference in the dy- ∈ } base associated with Orb(x0). The so-called ω-limit namical behavior of the system on + and its com- D i set of x0 given by plement be appreciated. At any fixed point x of f def in + (or at its equivalent images in [x]) the dynam- ω(x ) = x X : ( n N)(n )(f nk (x ) x) D 0 { ∈ ∃ k ∈ k →∞ 0 → } ics eventually gets attached to the (equivalent) fixed = x X : ( N )( fR ) point, and the sequence of iterates converges graph- { ∈ ∀ ∈ Nx ∀ i ∈F Bx0 ically in Multi(X) to x (or its equivalent points). (fRi (x0) N = ) (55) i 6 ∅ } When x / +, however, the orbit A = f (x) i∈N j ∈ D i { } is simply the adherence\ set adh(f (x0)) of the se- is chaotic in the sense that (f (x)) is not asymp- j quence (f (x0))j∈N, see Eq. (A.39); hence Defini- totically periodic and not being attached to any tion A.1.11 of the filter-base associated with a se- particular point they wander about in the closed quence and Eqs. (A.16), (A.24), (A.31) and (A.34) chaotic set ω(x) = Der(A) containing A such that allow us to express ω(x0) more meaningfully as for any given point in the set, some subsequence of the chaotic orbit gets arbitrarily close to it. Such ω(x ) = Cl(f (x )) . (56) 0 Ri 0 sequences do not converge anywhere but only fre- N i\∈ quent every point of Der(A). Thus although in the It is clear from the second of Eqs. 55) that for limit of progressively larger iterations there is com- a continuous f and any x X, x ω(x0) im- plete uncertainty of the outcome of an experiment ∈ ∈ plies f(x) ω(x0) so that the entire orbit of x conducted at either of these two categories of ini- ∈ lies in ω(x0) whenever x does imply that the ω- tial points, whereas on + this is due to a random limit set is positively invariant; it is also closed be- choice from a multifunctionalD set of equally prob- cause the adherent set is a closed set according to able outputs as dictated by the specific conditions Theorem A.1.3. Hence x ω(x ) A ω(x ) under which the experiment was conducted at that 0 ∈ 0 ⇒ ⊆ 0 reduces the ω-limit set to the closure of A with- instant, on its complement the uncertainty is due out any isolated points, A Der(A). In terms to the chaotic behavior of the functional iterates ⊆ of Eq. (A.33) involving principal filters, Eq. (56) themselves. Nevertheless it must be clearly under- in this case may be expressed in the more trans- stood that this later behavior is entirely due to the j ∞ parent form ω(x0) = Cl(F ( f (x0) )) where multifunctional limits at the points which com- P { }j=0 D+ the principal filter ( f j (x ) ∞ ) at A consists pletely determine the behavior of the system on its F T 0 j=0 P { j }∞ complement. As an explicit illustration of this sit- of all supersets of A = f (x0) j=0, and ω(x0) represents the adherence set{ of the} principal filter at uation, recall that for the shift map 2x mod(1) the points are the rationals on [0, 1], and any ir- A, see the discussion following Theorem A.1.3. If D+ A represents a chaotic orbit under this condition, rational is represented by a non-terminating and then ω(x0) is sometimes known as a chaotic set non-repeating decimal so that almost all decimals [Alligood et al., 1997]; thus the chaotic orbit in- in [0, 1] in any base contain all possible sequences finitely often visits every member of its chaotic of any number of digits. For the logistic map, the set25 which is simply the ω-limit set of a chaotic situation is more complex, however. Here the on- orbit that is itself contained in its own limit set. set of chaos marking the end of the period dou- Clearly the chaotic set is positive invariant, and bling sequence at λ∗ = 3.5699456 is signaled by the from Theorem A.1.3 and its corollary it is also com- disappearance of all stable fixed points, Fig. 13(c), pact. Furthermore, if all (sub)sequences emanating with Fig. 13(a) being a demonstration of the sta- from points x0 in some neighborhood of the set con- ble limits for λ = 3.569 that show up as conver- verge to it, then ω(x0) is called a chaotic attractor, gence of the iterates to constant valued functions see [Alligood et al., 1997]. As common examples of (rather than as constant valued inverse functions) chaotic sets that are not attractors mention may be at stable fixed points, shown more emphatically in made of the tent map with a peak value larger than Fig. 12(a). What actually happens at λ∗ is shown in 1 at 0.5, and the logistic map with λ 4 again with Fig. 16(a) in the next subsection: the almost verti- a peak value at 0.5 exceeding 1. ≥ cal lines produced at a large, but finite, iterations i

25 i How does this happen for A = f (x ) i N that is not the constant sequence (x ) at a ﬁxed point? As i N increases, { 0 } ∈ 0 ∈ points are added to x , f(x ), . . . , f I (x ) not, as would be the case in a normal sequence, as a piled-up Cauchy tail, but as { 0 0 0 } points generally lying between those already present; recall a typical graph as of Fig. 9, for example. December 3, 2003 12:13 00851

3188 A. Sengupta

1 1 1 1

99 2 2 55 6 6 1 21 2 7 7 3 3 10 10 1 1

Stable 1-cycle,Stable 1-cycle,λ =2.95λ =2.95 Stable 2-cycle,Stable 2-cycle,λ =3.4λ =3.4 GraphicalGraplimhiticalatlim9001it at 9001 GraphicalGraplimhiticalat lim9001-9002it at 9001-9002

01010.2 0.2 0 0 0.7 0.7 11 (a) (b) (a) (b) (b)

1 1 1 1

9 9 5 5 1 1 2 2

6 6 10 10 8 8

StableSt4-cycle,able 4-cycle,λ =3λ.5=3.5 StableS8-cycle,table 8-cycle,λ =3.55λ =3.55 GrapGrhicalaphlimicalit limat 9001-9004it at 9001-9004 GraphGricalaplimhicalit atlim9001-9008it at 9001-9008 0 0 0.7 0.7 1 0 1 0 0.7 0.7 1 1 (c) (d)

Fig. 12. Fixed points and cycles of logistic map. The isolated fixed point of (b) yields two non-fixed points to which the iterates converge simultaneously in the sense that the generated sequence converges to one iff it converges to the other. This suggests that nonlinear dynamics of a system can lead to a situation in which sequences in a Hausdorff space may converge to more than one point. Since convergence depends on the topology (Corollary to Theorem A.1.5), this may be interpreted to mean that nonlinearity tends to modify the basic structure of a space. The sequence of points generated by the iterates of the map are marked on the y-axis of (a)–(c) in italics. The singletons x are ω-limit sets of the respective fixed point x and { } is generated by the constant sequence (x, x, . . .). Whereas in (a) this is the limit of every point in (0, 1), in the other cases these fixed points are isolated in the sense of Definition 2.3. The isolated points, however, give rise to sequences that converge to more than one point in the form of limit cycles1 as sho1wn in (b)–(d).

(the multifunctions are generated only in the limit- chaos therefore, λx(1 x) is chaotic for the values − ing sense of i and represent a boundary be- of λ > λ∗ that are shown in Fig. 16. We return to tween functional→ and∞ non-functional relations on a this case in the following subsection. set), decrease in magnitude with increasing itera- As an example of chaos in a noniterative sys- tions until they reduce to points. This gives rise to tem, we investigate the following question: While a (totally disconnected) Cantor set on the y-axis in maximality of non-injectiveness produced by an in- contrast with the connected intervals that the mul- creasing number of injective branches is necessary tifunctional limits at λ > λ∗ of Figs. 16(b)–16(d) for a family of functions to be chaotic, is this also produce. By our characterization Deﬁnition 4.1 of suﬃcient for the system to be chaotic? This is an

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Toward a Theory of Chaos 3189

.507 .507 1 1

.5 .5 Iterate 9000Iteratofe 9000logistofic lmogiapstic map 9000 iterat9000ionitseratonilooginsstonic lmogiapstic map λ . λ . λ . λ . .473 .473 Order atOrd=3er at569=3569.488 0.4880.1 0 0.1 at =3at569=3569 1 1 (a) (a) (b) (b) (a) (b)

.511 .511 1 1

Iterate 9000 of logistic map 9000 iterations on logistic map .493 .493 Iterate 9000 of logistic map 9000 iterations on logistic map ‘‘ .472 .472Edgeof”Edchaos”ge ofatchλaos”=λ∗at λ =.487λ∗ 01.4870.1 010.at1λ∗ =3at.5699456λ∗ =3.5699456 (c) (d) (c) (c) (d)(d)

.511 .511 1 1

1 1

.493 .493Iterate 9000Iteratofelogi9000sticofmlaogip stic map 9000 iterat9000ionsitoneratlogiionstsiconmlaogip stic map λ . λ . λ . λ . .472 .472 Chaos atCh=3aos 57at =357 .487 0 .0.4871 0 0.1 at =357at =357 1 1 (e) (e) (f) (f) (e) (f)

Fig. 13. Multifunctional and cobweb plots of λx(1 x). Comparison of the graphs for the three values of λ shown in (a)–(f) − illustrates how the dramatic changes in the character of the former are conspicuously absent in the conventional plots that display no perceptible distinction between the three cases.

1 1

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3190 A. Sengupta

i important question especially in the context of a functions (f )i∈N which may be verified by reference non-iterative family of functions where fixed points to Definition A.1.8, Theorem A.1.3 and the proofs are no longer relevant. of Theorems A.1.4 and A.1.5, together with the di- Consider the sequence of functions rected set Eq. (A.10) with direction (A.11). The ∞ sin(πnx) n=1. The graphs of the subsequence basin of attraction of the attractor is A1 because | sin(2n−1π| x) and of the sequence (tn(x)) on [0, 1] | | the graphical limit ( +, F ( +)) (G( +), +) of are qualitatively similar in that they both contain Definition 3.1 may bDe obtained,D as indicatedR Rabove, n−1 2 of their functional graphs each on a base of by a proper choice of sequencesS associated with n−1 n−1 ∞ n ∞ 1/2 . Thus both sin(2 πx) n=1 and (t (x))n=1 . Note that in the context of iterations of func- converge graphically| to the multifunction| [0,1] on A tions, the graphical limit ( +, y0) of the sequence the same set of points equivalent to 0. This is suf- n D (f (x)) denotes a stable fixed point x∗ with im- ficient for us to conclude that sin(2n−1πx) ∞ , n=1 age x∗ = f(x∗) = y0 to which iterations start- and hence sin(πnx) ∞ , is chaotic| on the infinite| n=1 ing at any point x + converge. The graphi- equivalent |set [0]. While| Fig. 9 was a comparison ∈ D cal limits (xi0, +) are generated with respect to of the first four iterates of the tent and absolute R the class xi∗ of points satisfying f(xi0) = xi∗, sine maps, Fig. 14 shows the “converged” graphical i = 0, 1, {2, . .}. equivalent to unstable fixed point limits after 17 iterations. x∗ := x0∗ to which inverse iterations starting at any initial point in must converge. Even though only 4.1. The chaotic attractor R+ x∗ is inverse stable, an equivalent class of graph- One of the most fascinating characteristics of chaos ically converged limit multis is produced at every in dynamical systems is the appearance of attrac- member of the class xi∗ [x∗], resulting in the far- tors the dynamics on which are chaotic. For a subset reaching consequence that∈every member of the class A of a topological space (X, ) such that (f(A)) is as significant as the parent fixed point x from U R ∗ is contained in A — in this section, unless otherwise which they were born in determining the dynam- stated to the contrary, f(A) will denote the graph ics of the evolving system. The point to remember and not the range (image) of f — which ensures about infinite intersections of a collection of sets that the iteration process can be carried out in A, having finite intersection property, as in Eq. (58), is let that this may very well be empty; recall, however, j that in a compact space this is guaranteed not to be fRi (A) = f (A) N so. In the general case, if core( ) = then is the j≥[i∈ A 6 ∅ A (57) principal filter at this core, and Atr(A1) by Eqs. (58) j = f (x) and (A.33) is the closure of this core, which in this N ! j≥[i∈ x[∈A case of topology being induced by the filterbase, is generate the filter-base F with Ai := fRi (A) F just the core itself. A1 by its very definition, is a pos- being decreasingly nested,B A A for all i∈ NB, itively invariant set as any sequence of graphs con- i+1 ⊆ i ∈ in accordance with Definition A.1.1. The existence verging to Atr(A1) must be eventually in A1: the of a maximal chain with a corresponding maxi- entire sequence therefore lies in A1. Clearly, from mal element as asssured by the Hausdorff Maximal Theorem A.3.1 and its corollary, the attractor is a Principle and Zorn’s Lemma respectively implies a positively invariant compact set. A typical attrac- nonempty core of . As in Sec. 3 following Defi- tor is illustrated by the derived sets in the second FB nition 3.3, we now identify the filterbase with the column of Fig. 22 which also illustrates that the set neighborhood base at f ∞ which allows us to define of functional relations are open in Multi(X); specifi- def cally functional–non-functional correspondences are Atr(A ) = adh( ) 1 FB neutral-selfish related as in Fig. 22, 3–2, with the (58) = Cl(Ai) attracting graphical limit of Eq. (58) forming the A ∈FB boundary of (finitely) many-to-one functions and i\ as the attractor of the set A1, where the last equal- the one-to-(finitely) many multifunctions. ity follows from Eqs. (59) and (20) and the closure Equation (58) is to be compared with the im- is wth respect to the topology induced by the neigh- age definition of an attractor [Stuart & Humphries, borhood filter base F . Clearly the attractor as de- 1996] where f(A) denotes the range and not the fined here is the graphicalB limit of the sequence of graph of f. Then Eq. (58) can be used to define a December 3, 2003 12:13 00851

Toward a Theory of Chaos 3191

1111

16 0 017th iter17thate ofitertenatet mofatenp t m.0008ap 0G.0008 0Graphofr|asin(ph2o16fπx|sin()| 2 πx.0008)| .0008

(a) (a) (b) (b)

Fig. 14. Similarity in the behavior of the graphs of (a) tent and (b) sin(216πx) maps at 17 iterations demonstrate chaoticity | | of the latter.

sequence of points x A and hence the subset be identified with the subset on the y-axis on k ∈ nk R+ def which the multifunctional limits G : + X of ω(A) = x X : ( n N)(n )( x A ) R → { ∈ ∃ k ∈ k → ∞ ∃ k ∈ nk graphical convergence are generated, with its basin nk (f (xk) x) of attraction being contained in the + associated → } with the injective branch of f that generatesD . In = x X : ( N )( A ) R+ { ∈ ∀ ∈ Nx ∀ i ∈ A summary it may be concluded that since definitions (N Ai = ) (59) (59) and (61) involve both the domain and range 6 ∅ } of f, a description of the attractor in terms of the \ as the corresponding attractor of A that satisfies an graph of f, like that of Eq. (58), is more pertinent equation formally similar to (58) with the difference and meaningful as it combines the requirements of that the filter-base is now in terms of the image both these equations. Thus, for example, as ω(A) is f(A) of A, which alloA ws the adherence expression not the function G( +), this attractor does not in- to take the particularly simple form clude the equivalenceR class of inverse stable points i ω(A) = Cl(f (A)) . (60) that may be associated with x∗, see for example i∈N Fig. 15. \ From Eq. (59), we may make the particularly The complimentary subset excluded from this def- nk simple choice of (xk) to satisfy f (x−k) = x so inition of ω(A), as compared to Atr(A1), that is that x = f −nk (x), where x [x ] := f −nk (x) required to complete the formalism is given by −k B −k −k is the element of the equivalence∈class of the inverse Eq. (61) below. Observe that the equation for ω(A) image of x corresponding to the injective branch f . is essentially Eq. (A.15), even though we prefer to B This choice is of special interest to us as it is the use the alternate form of Eq. (A.16) as this brings 1class that generates the G-function on in graph- out more clearly the frequenting nature of the1se- + ical convergence. This allows us to expressR ω(A) as quence. The basin of attraction −nk B (A) = x A : ω(x) Atr(A) ω(A) = x X : ( nk N)(nk )(fB (x) f { ∈ ⊆ } { ∈ ∃ ∈ → ∞ = x A : ( nk N)(nk ) (61) = x converges in (X, )) ; (62) { ∈ ∃ ∈ → ∞ −k U } (f nk (x) x∗ ω(A)) → ∈ note that the x−k of this equation and the xk of of the attractor is the smallest subset of X in which Eq. (59) are, in general, quite different points. sequences generated by f must eventually lie in or- A simple illustrative example of the construc- der to adhere at ω(A). Comparison of Eqs. (62) tion of ω(A) for the positive injective branch of with (33) and (61) with (32) show that ω(A) can the homeomorphism (4x2 1)/3, 1 x 1, is − − ≤ ≤ December 3, 2003 12:13 00851

3192 A. Sengupta

4x2 − 1 f = 3 0.8 x−1

fB 0.6

x 0.4 2 fB

0.2 3 fB

1 −0.8−0.60−0.4 −0.20.20.4 0.60.8

5 4 3 2 −0.25 4 3 2 1 −0.4 x1x2 x−1x−3

Fig. 15. The attractor for f(x) = (4x2 1)/3, for 1 x 1. The converging sequences are denoted by arrows on the − − ≤ ≤ right, and (xk) are chosen according to the construction shown. This example demonstrates how although A f(A), where i ⊆ A = [0, 1] is the domain of the positive injective branch of f, the succeeding images (f (A))i≥1 satisfy the required restriction for iteration, and A in the discussion above can be taken to be f(A); this is permitted as only a ﬁnite number of iterates is thereby discarded. It is straightforward to verify that Atr(A ) = ( 1, [ 0.25, 1]) (( 1, 1), 0.25) (1, [ 0.25, 1]) with 1 − − − − − F (x) = 0.25 on − = ( 1, 1) = + and G(y) = 1, and 1 on − = [ 0.25, 1] = +. By comparison, ω(A) from either − D − D − R − SR S its deﬁnition Eq. (59) or from the equivalent intersection expression Eq. (60), is simply the closed interval + = [ 0.25, 1]. R − The italicized iterate numbers on the graphs show how the oscillations die out with increasing iterations from x = 1 and approach 0.25 in all neighborhoods of 0. −

shown in Fig. 15, where the arrow-heads denote the quirements of an attractor to lead to the concept of converging sequences f ni (x ) x and f ni−m(x ) a chaotic attractor to be that on which the dynam- i → i → x−m which proves invariance of ω(A) for a homeo- ics is chaotic in the sense of Definitions 4.1. and 4.2. morphic f; here continuity of the function and its Hence inverse is explicitly required for invariance. Posi- tive invariance of a subset A of X implies that for Definition 4.3 (Chaotic Attractor). Let A be a n any n N and x A, f (x) = yn A, while positively invariant subset of X. The attractor negative∈ invariance ∈assures that for an∈y y A, Atr(A) is chaotic on A if there is sensitive depen- −n ∈ f (y) = x−n A. Invariance of A in both the dence on initial conditions for all x A. The sensi- forward and bac∈kward directions therefore means tive dependence manifests itself as ∈multifunctional that for any y A and n N, there exists a x A graphical limits for all x + and as chaotic orbits n ∈ ∈ ∈ 1 ∈ D such that f (x) = y. In interpreting this figure, it when x +. may be useful to recall from Definition 4.1 that an 6∈ D increasing number of injective branches of f is a The picture of chaotic attractors that emerge necessary, but not sufficient, condition for the oc- from the foregoing discussions and our characteri- currence of chaos; thus in Figs. 12(a) and 15, in- zation of chaos of Definition 4.1 is that it it is a creasing noninjectivity of f leads to constant valued subset of X that is simultaneously “spiked” multi- limit functions over a connected + in a manner functional on the y-axis and consists of a dense col- similar to that associated with theDclassical Gibb’s lection of singleton domains of attraction on the x- phenomenon in the theory of Fourier series. axis. This is illustrated in Fig. 16 which shows some Graphical convergence of an increasingly non- typical chaotic attractors. The first four diagrams linear family of functions implied by its increasing (a)–(d) are for the logistic map with (b)–(d) show- non-injectivity may now be combined with the re- ing the 4-, 2- and 1-piece attractors for λ = 3.575, December 3, 2003 12:13 00851

Toward a Theory of Chaos 3193

3.66, and 3.8, respectively that are in qualitative significant as λ = λ∗ marks the boundary between agreement with the standard bifurcation diagram the nonchaotic region for λ < λ∗ and the chaotic for reproduced in (e). Figures 16(b)–16(d) have the ad- λ > λ∗ (this is to be understood as being suitably vantage of clearly demonstrating how the attractors modified by the appearance of the nonchaotic win- are formed by considering the graphically converged dows for some specific intervals in λ > λ∗). At λ∗ limit as the object of study unlike in (e) which shows the generated fractal Cantor set Λ is an attractor the values of the 501–1001th iterates of x0 = 1/2 as as it attracts almost every initial point x0 so that n n a function of λ. The difference in (a) and (b) for a the successive images x = f (x0) converge toward change of λ from λ > λ∗ = 3.5699456 to 3.575 is the Cantor set Λ. In (f) the chaotic attractors for

1 1 1 1

λ=3.λ5699456=3.5699456 λ =3.575λ =3.575 IteratesIter=ates2001=−20012004− 2004 IteratesIter=ates2001=−20012004− 2004 0 0 1010 1 1 (b) (a)(a) (a) (b)

111 1

λ =3.λ66=3.66 λ =3.8λ =3.8 IteratesIter=ates2001=−20012004− 2004 IteratesIter=ates2001=−20012002− 2002 01010 0 1 1 (d) (c)(c) (c) (d) Fig. 16. Chaotic attractors for diﬀerent values of λ. For the logistic map the usual bifurcation (e) shows the chaotic attractors for λ > λ∗ = 3.5699456, while (a)–(d) display the graphical limits for four values of λ chosen for the Cantor set and 4,- 2-, and 1-piece attractors, respectively. In (f) the attractor [0, 1] (where the dotted lines represent odd iterates and the solid lines even iterates of f) disappear if f is reﬂected about the x-axis. The function ff (x) is given by

2(1 + x)/3 0 x < 1/2 ff (x) = ≤ 2(1 x) 1/2 x 1 . ( − ≤ ≤

1 1

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3194 A. Sengupta

1 1 11

λ4=3.λ4494=3.449

0000λ4λλ4∗λ∗44First 1F2irstitera12teisterates 11 (e)(e) (e) (f) (f(f)) Fig. 16. (Continued )

the piecewise continuous function on [0, 1] on ordered sets, just as the role of the choice of 2(1 + x) 1 an appropriate problem-dependent basis was high- , 0 x < lighted at the end of Sec. 2. Chaos as manifest in its 3 ≤ 2 ff(x) =  attractors is a direct consequence of the increasing  1  2(1 x), x 1 , nonlinearity of the map with increasing iteration; − 2 ≤ ≤ we reemphasize that this is only a necessary condi-  is [0, 1] where thedotted lines represent odd iterates tion so that the increasing nonlinearities of Figs. 12 and the full lines even iterates of f; here the attrac- and 15 eventually lead to stable states and not to tor disappears if the function is reflected about the chaotic instability. Under the right conditions as x-axis. enunciated following Fig. 10, chaos appears to be the natural outcome of the difference in the behav- 4.2. Why chaos? A preliminary ior of a function f and its inverse f − under their inquiry successive applications. Thus f = ff −f allows f The question as to why a natural system should to take advantage of its multi-inverse to generate evolve chaotically is both interesting and relevant, all possible equivalence classes that are available, a − − − and this section attempts to advance a plausible an- feature not accessible to f = f ff . As we have swer to this inquiry that is based on the connection seen in the foregoing, equivalence classes of fixed between topology and convergence contained in the points, stable and unstable, are of defining signif- Corollary to Theorem A.1.5. Open sets are group- icance in determining the ultimate behavior of an ings of elements that govern convergence of nets evolving dynamical system and as the eventual (as and filters, because the required property of being also frequent) charcter of a filter or net in a set either eventually or frequently in (open) neighbor- is dictated by open neighborhoods of points of the hoods of a point determines the eventual behavior set, it is postulated that chaoticity on a set X leads of the net; recall in this connection the un1usual 1con- to a reformulation of the open sets of X to equiv- vergence characteristics in cofinite and cocountable alence classes generated by the evolving map f, see spaces. Conversely for a given convergence charac- Example 2.4(3). Such a redefinition of open sets of teristic of a class of nets, it is possible to infer the equivalence clases allow the evolving system to tem- topology of the space that is responsible for this porally access an ever increasing number of states convergence, and it is this point of view that we even though the equivalent fixed points are not fixed adopt here to investigate the question of this sub- under iterations of f except for the parent of the section: recall that our Definitions 4.1 and 4.2 were class, and can be considered to be the governing based on purely algebraic set-theoretic arguments criterion for the cooperative or collective behavior December 3, 2003 12:13 00851

Toward a Theory of Chaos 3195

of the system. The predominance of the role of f − to points x = y X then x y: x is of course 6 ∈ ∼ in f = ff −f in generating the equivalence classes equivalent to itself while x, y, z are equivalent to (that is exploiting the many-to-one character) of f each other iff they are simultaneously in every open is reflected as limit multis for f (i.e. constant f − set in which the net may eventually belong. This on ) in f − = f −ff −; this interpretation of the hall-mark of chaos can be appreciated in terms of R+ dynamics of chaos is meaningful as graphical con- a necessary obliteration of any separation property vergence leading to chaos is a result of pointwise bi- that the space might have originally possessed, see convergence of the sequence of iterates of the func- property (H3) in Appendix A.3. We reemphasize tions generated by f. But as f is a noninjective that a set in this chaotic context is required to act function on X possessing the property of increasing in a dual capacity depending on whether it carries nonlinearity in the form of increasing noninjectivity the initial or final topology under . M with iteration, various cycles of disjoint equivalence This preliminary inquiry into the nature of classes are generated under iteration, see for exam- chaos is concluded in the final section of this paper. ple Fig. 9(a) for the tent map. A reference to Fig. shows that the basic set XB, for a finite number n of 5. Graphical Convergence Works iterations of f, contains the parent of each of these open equivalent sets in the domain of f, with the We present in this section some real evidence in support of our hypothesis of graphical conver- topology on XB being the corresponding p-images of these disjoint saturated open sets of the domain. gence of functions in Multi(X, Y ). The example is In the limit of infinite iterations of f leading to the taken from neutron transport theory, and concerns multifunction (this is the f ∞ of Sec. 4.1), the the discretized spectral approximation [Sengupta, M generated open sets constitute a basis for a topol- 1988, 1995] of Case’s singular eigenfunction solution of the monoenergetic neutron transport equa- ogy on (f) and the basis for the topology of (f) D R tion, [Case & Zweifel, 1967]. The neutron transport are the corresponding -images of these equivalent M equation is a linear form of the Boltzmann equation classes. It is our contention that the motive force be- that is obtained as follows. Consider the neutron- hind evolution toward a chaos, as defined by Defini- moderator system as a mixture of two species of tion 4.1, is the drive toward a state of the dynamical gases each of which satisfies a Boltzmann equation system that supports ininality of the limit multi ; M of the type see Appendix A.2 with the discussions on Fig. and Eq. (26) in Sec. 2. In the limit of infinite iterations ∂ + v f (r, v, t) therefore, the open sets of the range (f) X are ∂t i · ∇ i R ⊆ the multi images that graphical convergence gener- 0 0 0 0 ates at each of these inverse-stable fixed points. X = dv dv1 dv Wij(vi v ; v1 v ) 1 → → 1 therefore has two topologies imposed on it by the Z Z Z Xj f (r, v0, t)f (r, v0 , t) f (r, v, t)f (r, v , t) dynamics of f: the first of equivalence classes gen- { i j 1 − i j 1 } erated by the limit multi in the domain of f and M where the second as -images of these classes in the range M 0 0 0 0 of f. Quite clearly these two topologies need not be Wij(vi v ; v1 v1) = v v1 σij(v v , v1 v1) the same; their intersection therefore can be defined → → | − | − − to be the chaotic topology on X associated with the σij being the cross-section of interaction between chaotic map f on X. Neighborhoods of points in this species i and j. Denote neutrons by subscript 1 and topology cannot be arbitrarily small as they consist the background moderator with which the neutrons of all members of the equivalence class to which interact by 2, and make the assumptions that

any element belongs; hence a sequence converging (i) The neutron density f1 is much less compared to any of these elements necessarily converges to with that of the moderator f2 so that the terms all of them, and the eventual objective of chaotic f1f1 and f1f2 may be neglected in the neutron and dynamics is to generate a topology in X with re- moderator equations, respectively.

spect to which elements of the set can be grouped (ii) The moderator distribution f2 is not aﬀected together in as large equivalence classes as possible by the neutrons. This decouples the neutron and in the sense that if a net converges simultaneously moderator equations and leads to an equilibrium December 3, 2003 12:13 00851

3196 A. Sengupta

Maxwellian fM for the moderator while the neu- the continuous spectrum of µ. This distinction be- trons are described by the linear equation tween the nature of the inverses depending on the ∂ relative values of µ and ν suggests a wider “non- + v f(r, v, t) function” space in which to look for the solutions of ∂t · ∇ operator equations, and in keeping with the philos- 0 0 0 0 ophy embodied in Fig. of treating inverse prob- = dv dv1 dv W12(v v ; v1 v ) 1 → → 1 lems in the space of multifunctions, we consider Z Z Z R f(r, v0, t)f (v0 ) f(r, v, t)f (v ) ) all ν Multi(V (µ), )) satisfying Eq. (63) to be { M 1 − M 1 } eigenfunctionsF ∈ of µ for the corresponding eigenvalue This is now put in the standard form of the neutron ν, leading to the following multifunctional solution transport equation [Williams, 1971] of (63) 1 ∂ + Ω v + (E) Φ(r, E, Ωˆ, t) (V (µ), 0) if ν / V (µ) v ∂t · S ν (µ) = ∈ F ( (V (µ) ν, 0) (ν, R)) if ν V (µ) , − ∈ = dΩ0 dE0 (r, E0 E; Ωˆ 0 Ω)Φ(ˆ r, E0, Ωˆ 0, t). where V (µ) ν is used asSa shorthand for the inter- S → · − Z Z val V (µ) with ν deleted. Rewriting the eigenvalue 2 ˆ equation (63) as µ ( (µ)) = 0 and comparing this where E = mv /2 is the energy and Ω the direc- ν Fν tion of motion of the neutrons. The steady state, with Fig. , allows us to draw the correspondences monoenergetic form of this equation is Eq. (A.53) f µ ⇔ ν ∂Φ(x, µ) c 1 X and Y Multi(V (µ), R) : µ + Φ(x, µ) = Φ(x, µ0)dµ0 , ⇔ {Fν ∈ ∂x 2 −1 ν (µν) Z F ∈ D } (64) 0 < c < 1, 1 µ 1 f(X) 0 : 0 Y − ≤ ≤ ⇔ { ∈ } and its singular eigenfunction solution for x X 0 : 0 X ∈ B ⇔ { ∈ } ( , ) is given by Eq. (A.56) − − −∞ ∞ f µν . −x/ν0 ⇔ Φ(x, µ) = a(ν0)e φ(µ, ν0) Thus a multifunction in X is equivalent to 0 in XB x/ν0 + a( ν0)e φ( ν0, µ) under the linear map µ , and we show below that − − ν 1 this multifunction is in fact the Dirac delta “func- −x/ν + a(ν)e φ(µ, ν)dν ; tion” δν (µ), usually written as δ(µ ν). This sug- −1 gests that in Multi(V (µ), R), every−ν V (µ) is in Z ∈ see Appendix A.4 for an introductory review of the point spectrum of µ, so that discontinuous func- Case’s solution of the one-speed neutron transport tions that are pointwise limits of functions in func- equation. tion space can be replaced by graphically converged The term “eigenfunction” is motivated by the multifunctions in the space of multifunctions. Com- following considerations. Consider the eigenvalue pleting the equivalence class of 0 in Fig. , gives the equation multifunctional solution of Eq. (63). From a comparison of the definition of ill- (µ ν) (µ) = 0, µ V (µ), ν R (63) − Fν ∈ ∈ posedness (Sec. 2) and the spectrum (Table 1), it is in the space of multifunctions Multi(V (µ), ( , clear that λ(x) = y is ill-posed iff )), where µ is in either of the intervals [ −∞1, 1] L (1) not injective λ P σ( ), which corre- ∞or [0, 1] depending on whether the given b−ound- λ λ Lsponds to the first⇔row∈of TableL 1. ary conditions for Eq. (A.53) is full-range or half (2) not surjective the values of λ correspond range. If we are looking only for functional solu- λ Lto the second and⇔third columns of Table 1. tions of Eq. (63), then the unique function that (3) is bijective but not open λ is either in satisfies this equation for all possible µ V (Fµ) and λ LCσ( ) or Rσ( ) corresponding⇔ to the second ν R V (µ) is (µ) = 0 which means,∈ according λ λ ν row Lof Table 1.L to∈Table− 1, thatFthe point spectrum of µ is empty and (µ ν)−1 exists for all ν. When ν V (µ), how- We verify in the three steps below that X = − ∈ ever, this inverse is not continuous and we show L1[ 1, 1] of integrable functions, ν V (µ) = [ 1, 1] below that in Map(V (µ), 0), ν V (µ) belongs to belongs− to the continuous spectrum∈ of µ. − ∈ December 3, 2003 12:13 00851

Toward a Theory of Chaos 3197

Table 1. Spectrum of linear operator Map(X). Here := λ satisﬁes L ∈ Lλ L− the equation (x) = 0, with the resolvent set ρ( ) of consisting of all those Lλ L L complex numbers λ for which −1 exists as a continuous operator with dense Lλ domain. Any value of λ for which this is not true is in the spectrum σ( ) L of , that is further subdivided into three disjoint components of the point, L continuous and residual spectra according to the criteria shown in the table.

( ) R Lλ −1 = X Cl( ) = X Cl( ) = X Lλ Lλ R R R 6 Not injective P σ( ) P σ( ) P σ( ) · · · L L L Not continuous Cσ( ) Cσ( ) Rσ( ) Injective L L L Continuous ρ( ) ρ( ) Rσ( ) L L L

(a) (µν) is dense, but not equal to L1. The set Nevertheless although the net of functions R −1 of functions g(µ) L1 such that µν g L1 1 ∈ ∈ δ (µ) = cannot be the whole of L1. Thus, for example, νε tan−1(1 + ν)/ε + tan−1(1 ν)/ε the piecewise constant function g = const = 0 − 6 ε on µ ν δ > 0 and 0 otherwise is in L1 , ε > 0 | − | ≤ −1 × (µ ν)2 + ε2 but not in (µν) as µν g L1. Nevertheless − R 6∈ 1 for any g L1, we may choose the sequence of ∈ is in the domain of µν because −1 δνε(µ)dµ = 1 functions for all ε > 0, R 1 0, if µ ν 1/n lim µ ν δνε(µ)dµ = 0 ε→0 −1 | − | gn(µ) = | − | ≤ Z ( g(µ), otherwise implying that (µ ν)−1 is unbounded. Taken together,− (a) and (b) show that func- in (µ ) to be eventually in every neighbor- tional solutions of Eq. (63) lead to state 2–2 in R ν hood of g in the sense that lim 1 g Table 1; hence ν [ 1, 1] = Cσ(µ). n→∞ −1 | − ∈ − gn = 0. (c) The two integral constraints in (b) also mean | −1 R that ν Cσ(µ) is a generalized eigenvalue (b) The inverse (µ ν) exists but is not contin- ∈ uous. The inverse− exists because, as noted ear- of µ which justiﬁes calling the graphical limit G lier, 0 is the only functional solution of Eq. (63). δ (µ) δ (µ) a generalized, or singular, eigen- νε → ν

20 20 −32 32

−0.5 0.5 0.5 0 0

−0.5 −0.5 0 0 0.5 −32 −32 (a) (a) (b) (b) (a) (b)

2 2 Fig. 17. Graphical convergence of: (a) Poisson kernel δε(x) = ε/π(x + ε ) and (b) conjugate Poisson kernel Pε(x) = x/(x2 + ε2) to the Dirac delta and principal value, respectively; the graphs, each for a deﬁnite ε-value, converges to the respective limits as ε 0. →

1 1 December 3, 2003 12:13 00851

3198 A. Sengupta

function, see Fig. 17 which clearly indicates the with 26 convergence of the net of functions. 1 dµ 1 π = ε = 2 tan−1 ε→0 π . ε µ2 + ε2 ε −→ From the fact that the solution Eq. (A.56) of Z−1 the transport equation contains an integral involv- These discretized equations should be compared ing the multifunction φ(µ, ν), we may draw an in- with the corresponding exact ones of Appendix A.4. teresting physical interpretation. As the multi ap- We shall see that the net of functions (65) con- pears everywhere on V (µ) (i.e. there are no chaotic verges graphically to the multifunction Eq. (A.55) orbits but only the multifunctions that produce as ε 0. → them), we have here a situation typical of maximal In the discretized spectral approximation, ill-posedness characteristic of chaos: note that both the singular eigenfunction φ(µ, ν) is replaced by the functions comprising φ (µ, ν) are non-injective. φε(µ, ν), ε 0, with the integral in ν being replaced ε → As the solution (A.56) involves an integral over all by an appropriate sum. The solution Eq. (A.58) of ν V (µ), the singular eigenfunctions — that col- the physically interesting half-space x 0 problem ≥ lectiv∈ ely may be regarded as representing a chaotic then reduces to [Sengupta, 1988, 1995] substate of the system represented by the solution of Φ (x, µ) = a(ν )e−x/ν0 φ(µ, ν ) the neutron transport equation — combine with the ε 0 0 N functional components φ( ν , µ) to produce the 0 + a(ν )e−x/νi φ (µ, ν ) µ [0, 1] well-defined, non-chaotic, experimental end result i ε i ∈ of the neutron flux Φ(x, µ). Xi=1 The solution (A.56) is obtained by assuming (66) −x/ν N Φ(x, µ) = e φ(µ, ν) to get the equation for where the nodes νi i=1 are chosen suitably. This φ(µ, ν) to be (µ ν)φ(µ, ν) = cν/2 with the nor- discretized spectral{ }approximation to Case’s so- 1 − − −1 malization −1 φ(µ, ν) = 1. As µν is not invert- lution has given surprisingly accurate numerical ible in Multi(V (µ), R) and µνB : XB f(X) does results for a set of properly chosen nodes when not exist, theR alternate approach of regularization→ compared with exact calculations. Because of its was adopted in [Sengupta, 1988, 1995] to rewrite involved nature [Case & Zweifel, 1967], the exact µ φ(µ, ν) = cν/2 as µ φ (µ, ν) = cν/2 with calculations are basically numerical which leads to ν − νε ε − µνε := µ (ν + iε) being a net of bijective func- nonlinear integral equations as part of the solutions for ε−> 0; this is a consequence of the fact tion procedure. To appreciate the enormous com- that for the multiplication operator every non-real plexity of the exact treatment of the half-space λ belongs to the resolvent set of the operator. The problem, we recall that the complete set of eigen- family of solutions of the latter equation is given by functions φ(µ, ν ), φ(µ, ν) are orthogo- { 0 { }ν∈[0,1]} [Sengupta 1988, 1995] nal with respect to the half-range weight function W (µ) of half-range theory, Eq. (A.61), that is ex- cν ν µ λ (ν) ε pressed only in terms of solution of the nonlin- φ (ν, µ) = − + ε ε 2 (µ ν)2 + ε2 π (µ ν)2 + ε2 ear integral equation Eq. (A.62). The solution of − ε − (65) a half-space problem then evaluates the coefficients a(ν ), a(ν) from the appropriate half range { 0 ν∈[0, 1]} 1 (that is 0 µ 1) orthogonality integrals satisfied where the required normalization −1 φε(ν, µ) = 1 ≤ ≤ gives by the eigenfunctions φ(µ, ν0), φ(µ, ν) ν∈[0, 1] R with respect to the weigh{t W (µ), see{ Appendix} A.4} πε for the necessary details of the half-space problem λ (ν) = ε tan−1(1 + ν)/ε + tan−1(1 ν)/ε in neutron transport theory. − As may be appreciated from this brief introduc- cν (1 + ν)2 + ε2 1 ln tion, solutions to half-space problems are not sim- × − 4 (1 ν)2 + ε2 − ple and actual numerical computations must rely a ε→0 πλ(ν) great deal on tabulated values of the X-function. −→

26The technical deﬁnition of a generalized eigenvalue is as follows. Let be a linear operator such that there exists in the L domain of a sequence of elements (xn) with xn = 1 for all n. If limn→∞ ( λ)xn = 0 for some λ C, then this λ is L k k k L − k ∈ a generalized eigenvalue of , the corresponding eigenfunction x∞ being a generalized eigenfunction. L December 3, 2003 12:13 00851

Toward a Theory of Chaos 3199

Self-consistent calculations of sample benchmark nature of the exact theory, it is our contention that problems performed by the discretized spectral ap- the remarkable accuracy of these basic data, some proximation in a full-range adaption of the half- of which is reproduced in Table 2, is due to the range problem described below that generate all graphical convergence of the net of functions necessary data, independent of numerical tables, G with the quadrature nodes ν N taken at the φε(µ, ν) φ(µ, ν) { i}i=1 → zero Legendre polynomials show that the full range shown in Fig. 18; here ε = 1/πN so that ε 0 formulation of this approximation [Sengupta, 1988, as N . By this convergence, the delta→ 1995] can give very accurate results not only of inte- function→and∞principal values in [ 1, 1] are the grated quantities like the flux Φ and leakage of par- multifunctions ([ 1, 0), 0) (0, [0, − ) ((0, 1], 0) − ∞ ticles out of the half space, but of also basic “raw” and 1/x x∈[−1, 0) (0, ( , )) 1/x x∈(0, 1] data like the extrapolated end point respectiv{ ely.}Tables 2 and 3,−∞Staken∞from{S[Sengupta,} 1 2 1988] and [Sengupta,S1995], show respSectively the cν0 ν cν ν0 + ν z0 = 1 + 2 ln dν extrapolated end point and X-function by the 4 N(ν) 1 ν ν0 ν Z0 − − full-range adaption of the discretized spectral ap- (67) proximation for two different half-range problems and of the X-function itself. Given the involved denoted as Problems A and B defined as

c =0c .=03 .3 c=0c.=09 .9

c=0c.=03 .3 c=0c.=09 .9 N=N1000=1000 N =N1000= 1000

(a)(a) (a) (b) (b(b))

Fig. 18. Rational function approximations φε(µ, ν) of the singular eigenfunction φ(µ, ν) at four diﬀerent values of ν. N = 1000 denotes the “converged” multifunction φ, with the peaks at the speciﬁc ν-values chosen.

1 1 December 3, 2003 12:13 00851

3200 A. Sengupta

P roblem A Equation : µΦ + Φ = (c/2) 1 Φ(x, µ0)dµ0, x 0 x −1 ≥ Boundary condition : Φ(0, µ) = 0 for µ 0 R ≥ Asymptotic condition : Φ e−x/ν0 φ(µ, ν ) as x . → 0 → ∞ P roblem B Equation : µΦ + Φ = (c/2) 1 Φ(x, µ0)dµ0, x 0 x −1 ≥ Boundary condition : Φ(0, µ) = 1 for µ 0 R ≥ Asymptotic condition : Φ 0 as x . → → ∞

The full 1 µ 1 range form of the half of the full-range weight function µ as compared to 0 µ −1 range≤ discretized≤ spectral approxima- the half-range function W (µ), and the resulting sim- tion≤ replaces≤ the exact integral boundary condition plicity of the orthogonality relations that follow, see at x = 0 by a suitable quadrature sum over the val- Appendix A.4. The basic data of z0 and X( ν) ues of ν taken at the zeros of Legendre polynomials; are then completely generated self-consisten−tly thus the condition at x = 0 can be expressed as [Sengupta, 1988, 1995] by the discretized spectral approximation from the full-range adaption N N ψ(µ) = a(ν0)φ(µ, ν0) + a(νi)φε(µ, νi) , (68) b φ (µ, ν ) = ψ (µ) + ψ (µ) , i=1 i ε i + − (69) X i=0 µ [0, 1] , X µ [ 1, 1], ν 0 ∈ ∈ − i ≥ where ψ(µ) = Φ(0, µ) is the specified incoming radiation incident on the boundary from the left, Table 2. Extrapolated end-point z0. and the half-range coefficients a(ν ), a(ν) 0 ν∈[0,1] cz0 are to be evaluated using the W -function{ } of Appendix A.4. We now exploit the relative sim- c N = 2 N = 6 N = 10 Exact plicity of the full-range calculations by replacing 0.2 0.78478 0.78478 0.78478 0.7851 Eq. (68) by Eq. (69) following, where the coefficients 0.4 0.72996 0.72996 0.72996 0.7305 N b(νi) i=0 are used to distinguish the full-range co- 0.6 0.71535 0.71536 0.71536 0.7155 efficien{ }ts from the half-range ones. The significance 0.8 0.71124 0.71124 0.71124 0.7113 of this change lies in the overwhelming simplicity 0.9 0.71060 0.71060 0.71061 0.7106

Table 3. X( ν) by the full-range method. − X( ν) − c N νi Problem A Problem B Exact

0.2133 0.8873091 0.8873091 0.887308 0.2 2 0.7887 0.5826001 0.5826001 0.582500 0.0338 1.3370163 1.3370163 1.337015 0.1694 1.0999831 1.0999831 1.099983 0.3807 0.8792321 0.8792321 0.879232 0.6 6 0.6193 0.7215240 0.7215240 0.721524 0.8306 0.6239109 0.6239109 0.623911 0.9662 0.5743556 0.5743556 0.574355 0.0130 1.5971784 1.5971784 1.597163 0.0674 1.4245314 1.4245314 1.424532 0.1603 1.2289940 1.2289940 1.228995 0.2833 1.0513750 1.0513750 1.051376 0.4255 0.9058140 0.9058410 0.905842 0.9 10 0.5744 0.7934295 0.7934295 0.793430 0.7167 0.7102823 0.7102823 0.710283 0.8397 0.6516836 0.6516836 0.651683 0.9325 0.6136514 0.6136514 0.613653 0.9870 0.5933988 0.5933988 0.593399 December 3, 2003 12:13 00851

Toward a Theory of Chaos 3201

of the discretized boundary condition Eq. (68), the required bj from these “negative” coefficients. where ψ+(µ) is by definition the incident flux ψ(µ) By equating these calculated bi with the exact half- for µ [0, 1] and 0 if µ [ 1, 0], while range expressions for a(ν) with respect to W (µ) as ∈ ∈ − outlined in Appendix A.4, it is possible to find nu- N − merical values of z0 and X( ν). Thus from the sec- bi φε(µ, νi) if µ [ 1, 0], νi 0 − N ψ (µ) =  ∈ − ≥ ond of Eq. (A.64), X( νi) i=1 is obtained with − i=0 { − }  X biB = aiB, i = 1, . . . , N, which is then substituted  0 if µ [0, 1] ∈ in the second of Eq. (A.63) with X( ν0) obtained  from a (ν ) according to Appendix A.4,− to compare is the emergen t angular distribution out of the A 0 medium. Equation (69) corresponds to the full- the respective aiA with the calculated biA from (71). range µ [ 1, 1], νi 0 form Finally the full-range coefficients of Problem A can ∈ − ≥ be used to obtain the X( ν) values from the sec- 1 ond of Eqs. (A.63) and compared− with the exact b(ν0)φ(µ, ν0) + b(ν)φ(µ, ν)dν 0 tabulated values as in Table 3. The tabulated val- Z 1 − − ues of cz0 from Eq. (67) show a consistent deviation = ψ+(µ) + b (ν0)φ(µ, ν0) + b (ν)φ(µ, ν)dν from our calculations of Problem A according to 0 Z a (ν ) = exp( 2z /ν ). Since the X( ν) values (70) A 0 − − 0 0 − of Problem A in Table 3 also need the same b0A as of boundary condition (A.59) with the first and sec- input that was used in obtaining z0, it is reasonable ond terms on the right having the same interpre- to conclude that the “exact” numerical integration tation as for Eq. (69). This full-range simulation of z0 is inaccurate to the extent displayed in Table 2. merely states that the solution (A.58) of Eq. (A.53) From these numerical experiments and Fig. 18 holds for all µ [ 1, 1], x 0, although it was ob- we may conclude that the continuous spectrum ∈ − ≥ tained, unlike in the regular full-range case, from [ 1, 1] of the position operator µ acts as the + the given radiation ψ(µ) incident on the bound- p−oints in generating the multifunctional Case sin-D ary at x = 0 over only half the interval µ [0, 1]. gular eigenfunction φ(µ, ν). Its rational approxima- ∈ To obtain the simulated full-range coefficients bi tion φε(µ, ν) in the context of the simple simulated − { } and bi of the half-range problem, we observe that full-range computations of the complex half-range there{ are} effectively only half the number of coef- exact theory of Appendix A.4, clearly demonstrates ficients as compared to a normal full-range prob- the utility of graphical convergence of sequence of lem because ν is now only over half the full inter- functions to multifunction. The totality of the mul- val. This allows us to generate two sets of equations tifunctions φ(µ, ν) for all ν in Figs. 18(c) and 18(d) from (70) by integrating with respect to µ [ 1, 1] endows the problem with the character of max- with ν in the half intervals [ 1, 0] and ∈[0,−1] to imal ill-posedness that is characteristic of chaos. obtain the two sets of coefficien− ts b− and b, re- This chaotic signature of the transport equation is spectively. Accordingly we get from Eq. (69) with however latent as the experimental output Φ(x, µ) j = 0, 1, . . . , N the sets of equations is well-behaved and regular. This important exam-

N ple shows how nature can use hidden and complex (+) − (−) chaotic substates to generate order through a pro- (ψ, φj−)µ = b (φi+, φj−) − i µ cess of superposition. Xi=0 N 1 (+) − (−) bj = (ψ, φj+)µ + bi (φi+, φj+)µ 6. Does Nature Support Nj ! Xi=0 Complexity? (71) The question of this section is basic in the light of where (φ )N represents (φ (µ, ν ))N , φ = the theory of chaos presented above as it may be j j=1 ε j j=1 0 φ(µ, ν0), the (+) ( ) superscripts are used to reformulated to the inquiry of what makes nature denote the integrations− with respect to µ [0, 1] support chaoticity in the form of increasing non- ∈ and µ [ 1, 0] respectively, and (f, g)µ denotes injectivity of an input–output system. It is the pur- the usual∈ −inner product in [ 1, 1] with respect pose of this section to exploit the connection be- to the full range weight µ. While− the ﬁrst set of tween spectral theory and the dynamics of chaos − N + 1 equations give bi , the second set produces that has been presented in the previous section. December 3, 2003 12:13 00851

3202 A. Sengupta

Since linear operators on finite dimensional spaces of functions whose images under the respective λ do not possess continuous or residual spectra, spec- converge to 0; recall the definition of footnote 26.L tral theory on infinite dimensional spaces essentially This observation generalizes to the dense extension involves limiting behavior to infinite dimensions of Multi|(X, Y ) of Map(X, Y ) as follows. If x + the familiar matrix eigenvalue–eigenvector problem. is not a fixed point of f(λ; x) = x, but there∈ Dis As always this means extensions, dense embeddings some n N such that f n(λ; x) = x, then the limit and completions of the finite dimensional problem n ∈generates a multifunction at x as was the that show up as generalized eigenvalues and eigen- case→with∞ the delta function in the previous section vectors. In its usual form, the goal of nonlinear spec- and the various other examples that we have seen tral theory consists [Appell et al., 2000] in the study so far in the earlier sections. −1 of Tλ for nonlinear operators Tλ that satisfy more One of the main goals of investigations on the general continuity conditions, like differentiability spectrum of nonlinear operators is to find a set in and Lipschitz continuity, than simple boundedness the complex plane that has the usual desirable prop- that is sufficient for linear operators. The following erties of the spectrum of a linear operator [Appell generalization of the concept of the spectrum of a et al., 2000]. In this case, the focus has been to find linear operator to the nonlinear case is suggestive. a suitable class of operators (X) with T (X), For a nonlinear map, λ need not appear only in a such that the resolvent set isCexpressed as ∈ C multiplying role, so that an eigenvalue equation can ρ(T ) = λ C : (T is 1 : 1)(Cl( (T ) = X) be written more generally as a fixed-point equation { ∈ λ R λ and (T −1 (X) on (T )) f(λ; x) = x λ ∈ C R λ } with the spectrum σ(T ) being defined as the com- with a fixed point corresponding to the eigenfunc- plement of this set. Among the classes (X) that tion of a linear operator and an “eigenvalue” being have been considered, beside spaces ofC continu- the value of λ for which this fixed point appears. ous functions C(X), are linear boundedness B(X), The correspondence of the residual and continu- Frechet differentiability C 1(X), Lipschitz continuous parts of the spectrum are, however, less trivial ity Lip(X), and Granas quasiboundedness Q(x), than for the point spectrum. This is seen from the where Lip(X) specifically takes into account the following two examples [Roman, 1975]. Let Aek = nonlinearity of T to define λkek, k = 1, 2, . . . be an eigenvalue equation with e being the jth unit vector. Then (A λ)e := T (x) T (y) j k T Lip = supx6=y k − k , ∞ − k k x y (λk λ)ek = 0 iff λ = λk so that λk k=1 P σ(A) − { } ∈ ∞ k − k (72) are the only eigenvalues of A. Consider now (λk) T (x) T (y) k=1 T = inf = k − k to be a sequence of real numbers that tends to | |lip x6=y x y a finite λ∗; for example, let A be a diagonal ma- k − k trix having 1/k as its diagonal entries. Then λ∗ that are plain generalizations of the corresponding norms of linear operators. Plots of f −(y) = x belongs to the continuous spectrum of A because λ R{ R∈ (A λ∗)e = (λ λ∗)e with λ λ∗ implies (f λ) : (f λ)x = y for the functions f : − k k − k k → D − − } → that (A λ∗)−1 is an unbounded linear operator 1 λx, x < 1 − and λ∗ a generalized eigenvalue of A. In the second f (x) = −(1 −λ)x, 1 −x 1 λa  − − ≤ ≤ example Aek = ek+1/(k + 1), it is not difficult to  1 λx, 1 < x verify that: (a) The point spectrum of A is empty, −  λx, x < 1 (b) The range of A is not dense because it does − −1 f (x) = (1 λ)x 1, 1 x 2 not contain e1, and (c) A is unbounded because λb  − − ≤ ≤ ∗ 1 λx, 2 < x Aek 0. Thus the generalized eigenvalue λ = 0 in  − this case→ belongs to the residual spectrum of A. In  λx x < 1 either case, limj→∞ ej is the corresponding general- f (x) = − λc √x 1 λx 1 x ized eigenvector that enlarges the trivial null space − − ≤ ∗ ( λ∗ ) of the generalized eigenvalue λ . In fact (x 1)2 + 1 λx 1 x 2 N L f (x) = in these two and the Dirac delta example of Sec. λd (1 − λ)x − otherwise≤ ≤ 5 of continuous and residual spectra, the general- − ized eigenfunctions arise as the limits of a sequence f (x) = tan−1(x) λx λe − 5 4 5 4 5 4 0 10 0 1 0 1010 11 2 − December 3, 2003 12:13 00851 1 2 −1 2 − 2 − 1 2 −1 1 −3 2 −1 3 −3 3 −3 3 −1 2 − 1 2 0 1 1 2 −4 4 0 1 0 1 − −44 44 Toward a The−ory5 of Chaos 3203 −1 (a) (b) −−5 −−1 5 5 4 110 3 5 (a(a)4) (b)(b) 10 3 10 3 210 5 4 00 1010 1 210 −1 20.5 100 0 10 − 3 2 −−1 −11 1 1 22 0.50.5 −0−011 33 2 − −3 33 1 2 −1 1 −1 2 −3 3 1 2 11 1.5 −0.5 − −45 0 11 1.5 −−0..5 2 1 −1 2 −−441.5 0 5 44 −−45 22 −−11 −45 0 1 3 3 −4 4 −3 3 −55 3 −−11 3 −1 −1 (a(a)) (b)(b) (c) (d) −−11 −−11 − (a)−1010 (b)33 (c) 5 1 (c)(c) (d)(d)10 10 (a) (b) 10 3 22101010 1010 0 1.5 1 0.5 0−0.5 −−11 20.5 1000 00 1.51.50.5 11 0.50.5 00−−00.5.5 2 33 −1 −1 0.50.5 1 22 0.5 0 −−11 − . 3 11 2 0 5 1 − − 1 −− .. 2 2 10 10 22 0055 − . 2 − −− 0 5 1.5 −−00..55 22 22 1010 1 1010 1 −−00..55 22 2 −−1 −4545 1 11 −1 2 1.5 −0.5 0 0.5 − −− − −34533 11 22 −0.500 .511.5 2 1 00 0.50.5 −− .. .. −11 −−11 00505000 515111.5.5 −3 3 − − (c)(c) (d)(d) 10 10 − − −−1010 −−10(e)10 (f) 1 11010 1010 (c) (d) (e)(e) (f(f)) (d) (e) (f) − . 2 10 10 00 1.51.5 11 0.50.5 00−005.5 0.5 2 22 0.5 Fig. 19. Inverses of fλ =2 f λ. The λ-values are−1shown on the graphs. 0 1.5 1 0.5 0−0.5 − −1 1 0.5 2 − . 2 −−001.55 − − 1 −2 1 2√ x λx,22 x −<1010 1 the complemen1010t of the resolvent set, is more diffi- − . − − −2 − − . −0.5 2 cult to find. Here the convenient characterization of 2 0 5 f (x) = (1 λ)x, 1 1 x 1 −2 2 −10 λf  − 1 10 − ≤ ≤ the resolvent of a continuous linear operator as the − . 2 −1 0 5  2√x 1 λx, 1 < x −1 22 1 0 0.5 set of all sufficiently large λ that satisfy λ > M is 0 − − 0.5 − . . −0050.5000 51.5111.5.5 | | taken−1from [App ell et al., 2000]2are shown in Fig. 19. of little significance as, unlike for a linear operator, 0 0.5 It is easy to verify that the Lipschitz and linear the non-existence of an inverse is not just due to −0.500−.10511.5 −−10 upper and lower10bounds of these maps are as in 10the set f −1(0) which happens to be the only way (e)(e) (f(f)) { } −10 Table 4. −10 a linear map can fail to be injective. Thus the map (e) The point sp(fectrum) defined2by defined piecewise as α + 2(1 α)x for 0 x < 1/2 2 − ≤ C and 2(1 x) for 1/2 x 1, with 0 < α < 1, is P σ(f) = λ : (f λ)x = 0 for some x = 0 − ≤ ≤ 2 { ∈ − 6 } not invertible on its range although f −(0) = 1. is the simplest to calculate. Because of the spe- { } Comparing Fig. 19 and Table 4, it is seen that in cial role played by the zero element 0 in generating cases (b)–(d), the intervals [ f , f ] are subsets the point spectrum in the linear case, the bounds | |b k kB m x x M x together with x = λx of the λ-values for which the respective maps are k k ≤ kL k ≤ k k L not injective; this is to be compared with (a), (e) imply Cl(P σ( )) = [ b, B] — where the subscripts denoteL the lokLkwer andkLkupper bounds in and (f) where the two sets are the same. Thus the Eq. (72) and which sometimes is taken to be a de- linear bounds are not good indicators of the unique- scriptor of the point spectrum of a nonlinear op- ness properties of solution of nonlinear equations for erator — as can be seen in Table 5 and verified which the Lipschitzian bounds are seen to be more from Fig. 19. The remainder of the spectrum, as appropriate. December 3, 2003 12:13 00851

3204 A. Sengupta

Table 4. Bounds on the functions of Fig. 19. Thus apart from multifunctions, λ σ(f) also generates functions on the boundary of∈functional Function f f f f | |b k kB | |lip k kLip and non-functional relations in Multi(X, ). While T fa 0 1 0 1 it is possible to classify the spectrum into point, continuous and residual subsets, as in the linear fb 0 1/2 0 1 case, it is more meaningful for nonlinear opera- fc 0 1/2 0 ∞ tors to consider λ as being either in the bound- f 2(√2 1) 0 2 d − ∞ ary spectrum Bdy(σ(f)) or in the interior spectrum fe 0 1 0 1 Int(σ(f)), depending on whether or not the mul- ff 0 1 0 1 tifunction f(λ; )− arises as the graphical limit of a net of functions· in either ρ(f) or Rσ(f). This is suggested by the spectra arising from the sec- Table 5. Lipschitzian and point spectra ond row of Table 1 (injective λ and discontinu- of the functions of Fig. 19. −1 L ous λ ) that lies sandwiched in the λ-plane be- tweenL the two components arising from the first Functions σ (f) P σ(f) Lip and third rows, see [Naylor & Sell, 1971, Sec. 6.6], for example. According to this simple scheme, the fa [0, 1] (0, 1] spectral set is a closed set with its boundary and f [0, 1] [0, 1/2] b interior belonging to Bdy(σ(f)) and Int(σ(f)), re- fc [0, ) [0, 1/2] ∞ spectively. Table 6 shows this division for the ex- f [0, 2] [2(√2 1), 1] d − amples in Fig. 19. Because 0 is no more significant fe [0, 1] (0, 1) than any other point in the domain of a nonlin- ff [0, 1] (0, 1) ear map in inducing non-injectivity, the division of the spectrum into the traditional sets would be as shown in Table 6; compare also with the conven- In view of the above, we may draw the follow- tional linear point spectrum of Table 5. In this non- ing conclusions. If we choose to work in the space linear classification, the point spectrum consists of of multifunctions Multi(X, ), with the topology any λ for which the inverse f(λ; )− is set-valued, T T of pointwise biconvergence, when all functional re- irrespective of whether this is produced· at 0 or not, lations are (multi)invertible on their ranges, we may while the continuous and residual spectra together make the following definition for the net of functions comprise the boundary spectrum. Thus a λ can be f(λ; x) satisfying f(λ; x) = x. both at the point and the continuous or residual spectra which need not be disjoint. The continuous Definition 6.1. Let f(λ; ) Multi(X, ) be a · ∈ T and residual spectra are included in the boundary function. The resolvent set of f is given by spectrum which may also contain parts of the point spectrum. ρ(f) = λ : (f(λ; )−1 Map(X, )) { · ∈ T (Cl( (f(λ; )) = X) , Example 6.1. To see how these concepts apply ∧ R · } to linear mappings, consider the equation (D and any λ not in ρ is in the spectrum of f. λ)y(x) = r(x) where D = d/dx is the differential−

Table 6. Nonlinear spectra of functions of Fig. 19. Compare the present point spectra with the usual linear spectra of Table 5.

Function Int(σ(f)) Bdy(σ(f)) P σ(f) Cσ(f) Rσ(f)

fa (0, 1) 0, 1 [0, 1] 1 0 { } { } { } f (0, 1) 0, 1 [0, 1] 1 0 b { } { } { } fc (0, ) 0 [0, ) 0 ∞ { } ∞ { } ∅ f (0, 2) 0, 2 (0, 2) 0, 2 d { } { } ∅ fe (0, 1) 0, 1 (0, 1) 1 0 { } { } { } f (0, 1) 0, 1 (0, 1) 0, 1 f { } { } ∅ December 3, 2003 12:13 00851

Toward a Theory of Chaos 3205

operator on L2[0, ), and let λ be real. For λ = 0, by the graphical convergence of a net of resolvent the unique solution∞of this equation in L2[0, 6 ), is functions while the multifunctions in the interior of ∞ the spectral set evolve graphically independent of x 0 eλx y(0) + e−λx r(x0)dx0 , λ < 0 the functions in the resolvent. The chaotic states 0 forming the boundary of the functional and multi- y(x) =  Z  ∞ 0 functional subsets of Multi(X) marks the transition  λx −λx 0 0  e y(0) e r(x )dx λ > 0 from the less efficient functional state to the more − x Z efficient multifunctional one.  showingthat for λ > 0 the inverse is functional so These arguments also suggest the following. that λ (0, ) belongs to the resolvent of D. How- ∈ ∞ The countably many outputs arising from the non- ever, when λ < 0, apart from the y = 0 solution injectivity of f(λ; ) corresponding to a given input (since we are dealing with a linear problem, only can be interpreted· to define complexity because in r = 0 is to be considered), eλx is also in L2[0, ) ∞ a nonlinear system each of these possibilities con- so that all such λ are in the point spectrum of D. stitute an experimental result in itself that may not For λ = 0 and r = 0, the two solutions are not nec- be combined in any definite predtermined manner. 6 ∞ essarily equal unless 0 r(x) = 0, so that the range This is in sharp contrast to linear systems where (D I) is a subspace of L2[0, ). To complete R − R ∞ a linear combination, governed by the initial con- the problem, it is possible to show [Naylor & Sell, ditions, always generate a unique end result; recall 1971] that 0 Cσ(D), see Example 2.2; hence the ∈ also the combination offered by the singular gen- continuous spectrum forms at the boundary of the eralized eigenfunctions of neutron transport the- functional solution for the resolvent-λ and the mul- ory. This multiplicity of possibilities that have no tifunctional solution for the point spectrum. With definite combinatorial property is the basis of the a slight variation of problem to y(0) = 0, all λ < 0 diversity of nature, and is possibly responsible for are in the resolvent set, while λ > 0 the inverse is Feigenbaum’s “historical prejudice”, [Feigenbaum, ∞ −λx bounded but must satisfy y(0) = 0 e r(x)dx = 1992], see Prelude 2. Thus order represented by 0 so that Cl( (D λ)) = L2[0, ). Hence λ > 0 be- the functional resolvent passes over to complexity R − 6 ∞ R long to the residual spectrum. The decomposition of the countably multifunctional interior spectrum of the complex λ-plane for these and some other via the uncountably multifunctional boundary that linear spectral problems taken from [Naylor & Sell, is a prerequisite for chaos. We may now strengthen 1971] is shown in Fig. 20. In all cases, the spectrum our hypothesis offered at the end of the previous due to the second row of Table 1 acts as a boundary section in terms of the examples of Figs. 19 and between that arising from the first and third rows, 20, that nature uses chaoticity as an intermediate which justifies our division of the spectrum for a step to the attainment of states that would other- nonlinear operator into the interior and boundary wise be inaccessible to it. Well-posedness of a sys- components. Compare with Example 2.2. tem is an extremely inefficient way of expressing a multitude of possibilities as this requires a different From the basic representation of the resolvent input for every possible output. Nature chooses to operator (1 f)−1 − express its myriad manifestations through the mul- 1 + f + f 2 + + f i + tifunctional route leading either to averaging as in · · · · · · the delta function case or to a countable set of well- in Multi(X), if the iterates of f converge to a multi- defined states, as in the examples of Fig. 19 corre- function for some λ, then that λ must be in the spec- sponding to the interior spectrum. Of course it is no trum of f, which means that the control parameter distraction that the multifunctional states arise re- − of a chaotic dynamical system is in its spectrum. Of spectively from fλ and fλ in these examples as f is course, the series can sum to a multi even otherwise: a function on X that is under the influence of both take fλ(x) to be identically x with λ = 1, for exam- f and its inverse. The functional resolvent is, for all ple, to get 1 P σ(f). A comparison of Tables 1 and practical purposes, only a tool in this structure of 5 reveal that∈in case (d), for example, 0 and 2 belong nature. −1 to the Lipschtiz spectrum because although fd is The equation f(x) = y is typically an input– not Lipschitz continuous, f = 2. It should also output system in which the inverse images at a func- k kLip be noted that the boundary between the functional tional value y0 represents a set of input parameters resolvent and multifunctional spectral set is formed leading to the same experimental output y0; this December 3, 2003 12:13 00851

3206 A. Sengupta

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Fig. 20. Spectra of some linear operators in the complex λ-plane. (a) Left shift (. . . , x , x , x , . . .) (. . . x , x , x , . . .) −1 0 1 7→ 0 1 2 on l ( , ), (b) Right shift (x , x , x , . . .) (0, x , x , . . .) on l [0, ), (c) Left shift (x , x , x , . . .) (x , x , x , . . .) 2 −∞ ∞ 0 1 2 7→ 0 1 2 ∞ 0 1 2 7→ 1 2 3 on l [0, ) of sequence spaces, and (d) d/dx on L ( , ) (e) d/dx on L [0, ) with y(0) = 0 and (f) d/dx on L [0, ). 2 ∞ 2 −∞ ∞ 2 ∞ 2 ∞ The residual spectrum in (b) and (e) arise from block (3–3) in Table 1, i.e. is one-to-one and −1 is bounded on non- Lλ Lλ dense domains in l [0, ) and L [0, ), respectively. The continuous spectrum therefore marks the boundary between two 2 ∞ 2 ∞ functional states, as in (a) and (e), now with dense and non-dense domains of the inverse operator.

is stability characterized by a complete insensitiv- is larger than a functional state represented by the ity of the output to changes in input. On the other singleton f(x ) . { 0 } hand, a continuous multifunction at x0 is a signal for a hypersensitivity to input because the output, Epilogue which is a definite experimental quantity, is a choice The most passionate advocates of the new science from the possibly infinite set f(x ) made by a { 0 } go so far as to say that twentieth-century science choice function which represents the experiment at will be remembered for just three things: relativity, that particular point in time. Since there will always quantum mechanics and chaos. Chaos, they contend, be finite differences in the experimental parameters has become the century’s third great revolution in when an experiment is repeated, the choice function the physical sciences. Like the first two revolutions, (that is the experimental output) will select a point chaos cuts away at the tenets of Newton’s physics. As from f(x ) that is representative of that experi- { 0 } one physicist put it: “Relativity eliminated the New- ment and which need not bear any definite relation tonian illusion of absolute space and time; quantum to the previous values; this is instability and sig- theory eliminated the Newtonian dream of a con- nals sensitivity to initial conditions. Such a state is trollable measurement process; and chaos eliminates of high entropy as the number of available states the Laplacian fantasy of deterministic predictability.” fC( f(x0) ) — where fC is the choice function — Of the three, the revolution in chaos applies to the { } 1 11 11 December 3, 2003 12:13 00851

Toward a Theory of Chaos 3207

universe we see and touch, to objects at human scale. Goldenfeld, N. & Kadanoff, L. P. [1999] “Simple lessons . . . There has long been a feeling, not always expressed from complexity,” Science 284, 87–89. openly, that theoretical physics has strayed far from Korevaar, J. [1968] Mathematical Methods, Vol. 1 (Aca- human intuition about the world. Whether this will demic Press, NY). prove to be fruitful heresy, or just plain heresy, no one Naylor, A. W. & Sell, G. R. [1971] Linear Operator knows. But some of those who thought that physics Theory is Engineering and Science Holt (Rinehart and might be working its way into a corner now look to Winston, NY). Peitgen, H.-O., Jurgens, H. & Saupe, D. [1992] Chaos chaos as a way out. and Fractals: New Frontiers of Science (Springer- [Gleick, 1987] Verlag, NY). Robinson, C. [1999] Dynamical Systems: Stability, Sym- bolic Dynamics and Chaos (CRC Press LLC, Boca Acknowledgments Raton). It is a pleasure to thank the referees for recom- Roman, P. [1975] Some Modern Mathematics for Physi- mending an enlarged Tutorial and Review revision cists and other Outsiders (Pergammon Press, NY). of the original submission Graphical Convergence, Sengupta, A. [1995a] “A discretized spectral approxima- Chaos and Complexity, and Professor Leon O. Chua tion in neutron transport theory. Some numerical con- for suggesting a pedagogically self-contained, jar- siderations,” J. Stat. Phys. 51, 657–676. Sengupta, A. [1995b] “Full range solution of half-space gonless version accessible to a wider audience for neutron transport problem,” ZAMP 46, 40–60. the present form of the paper. Financial assis- Sengupta, A. [1997] “Multifunction and generalized in- tance during the initial stages of this work from verse,” J. Inverse and Ill-Posed Problems 5, 265–285. the National Board for Higher Mathematics is also Sengupta, A. & Ray, G. G. [2000] “A multifunctional ex- acknowledged. tension of function spaces: Chaotic systems are maximally ill-posed,” J. Inverse and Ill-Posed Problems 8, 232–353. References Stuart, A. M. & Humphries, A. R. [1996] Dynamical Alligood, K. T., Sauer, T. D. & Yorke, J. A. [1997] Chaos, Systems and Numerical Analysis (Cambridge Univer- An Introduction to Dynamical Systems (Springer- sity Press). Verlag, NY). Tikhonov, A. N. & Arsenin, V. Y. [1977] Solutions of Ill- Appell, J., DePascale, E. & Vignoli, A. [2000] “A com- Posed Problems (V. H. Winston, Washington D.C.). parison of different spectra for nonlinear operators,” Waldrop, M. M. [1992] Complexity: The Emerging Sci- Nonlin. Anal. 40, 73–90. ence at the Edge of Order and Chaos (Simon and Brown, R. & Chua, L. O. [1996] “Clarifying chaos: Schuster). Examples and counterexamples,” Int. J. Bifurcation Willard, S. [1970] General Topology (Addison-Wesley, and Chaos 6, 219–249. Reading, MA). Campbell, S. I. & Mayer, C. D. [1979] Generalized Williams, M. M. R. [1971] Mathematical Methods of Inverses of Linear Transformations (Pitman Publish- Particle Transport Theory (Butterworths, London). ing Ltd., London). Case, K. M. & Zweifel, P. F. [1967] Linear Transport Theory (Addison-Wesley, MA). Appendix de Souza, H. G. [1997] “Opening address,” in The Im- This Appendix gives a brief overview of some as- pact of Chaos on Science and Society, eds. Grebogi, C. & Yorke, J. A. (United Nations University Press, pects of topology that are necessary for a proper Tokyo), pp. 384–386. understanding of the concepts introduced in this Devaney, R. L. [1989] An Introduction to Chaotic Dy- work. namical Systems (Addison-Wesley, CA). Falconer, K. [1990] Fractal Geometry (John Wiley, Chichester). A.1. Convergence in Topological Feigenbaum, M. [1992] “Foreword,” Chaos and Fractals: Spaces: Sequence, Net and New Frontiers of Science (Springer-Verlag, NY), Filter pp. 1–7. Gallagher, R. & Appenzeller, T. [1999] “Beyond reduc- In the theory of convergence in topological spaces, tionism,” Science 284, p. 79. countability plays an important role. To understand Gleick, J. [1987] Chaos: The Amazing Science of the Un- the significance of this concept, some preliminaries predictable (Viking, NY). are needed. December 3, 2003 12:13 00851

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The notion of a basis, or base, is a familiar one determines reciprocally the topology as in analysis: a base is a subcollection of a set which U may be used to construct, in a specified manner, any = U X : U = B . (A.4) element of the set. This simplifies the statement of U  ⊆  B∈ B a problem since a smaller number of elements of  [T  the base can be used to generate the larger class This means that the topology on Xcan be recon- of every element of the set. This philosophy finds structed from the base by taking all possible unions application in topological spaces as follows. of members of the base, and a collection of subsets Among the three properties (N1) (N3) of the of a set X is a topological base iff Eq. (A.4) of arbi- − neighborhood system of Tutorial 4, (N1) and trary unions of elements of generates a topology Nx T (N2) are basic in the sense that the resulting sub- on X. This topology, which Bis the coarsest (that is collection of can be used to generate the full the smallest) that contains , is obviously closed Nx T system by applying (N3); this basic neighborhood under finite intersections. SinceB the open set Int(N) system, or neighborhood (local ) base at x, is char- is a neighborhood of x whenever N is, Eq. (A.2) Bx acterized by and the definition Eq. (17) of x implies that the open neighborhood system of anyN point in a topo- (NB1) x belongs to each member B of x. B logical space is an example of a neighborhood base (NB2) The intersection of any two members of x at that point, an observation that has often led, to- contains another member of : B , B B x 1 2 x gether with Eq. (A.3), to the use of the term “neigh- ( B : B B B ). B ∈ B ⇒ ∃ ∈ Bx ⊆ 1 2 borhood” as a synonym for “non-empty open set”. The distinction between the two however is signifi- Formally, compareT Eq. (18), cant as neighborhoods need not necessarily be open

Definition A.1.1. A neighborhood (local) base x sets; thus while not necessary, it is clearly sufficient at x in a topological space (X, ) is a subcollectionB for the local basic sets B to be open in Eqs. (A.1) of the neighborhood system U having the prop- and (A.2). If Eq. (A.2) holds for every x N, then Nx ∈ erty that each N contains some member of the resulting x reduces to the topology induced by ∈ Nx N x. Thus the open basic neighborhood system x as given by B Eq. (18). B def x = B x : x B N for each N x In order to check if a collection of subsets B { ∈ N ∈ ⊆ ∈ N } TB (A.1) of X qualifies to be a basis, it is not necessary to verify properties (T1)–(T3) of Tutorial 4 for the determines the full neighborhood system class (A.4) generated by it because of the proper- = N X : x B N for some B ties (TB1) and (TB2) below whose strong affinity to Nx { ⊆ ∈ ⊆ ∈ Bx} (A.2) (NB1) and (NB2) is formalized in Theorem A.1.1. reciprocally as all supersets of the basic elements. Theorem A.1.1. A collection T of subsets of X is a topological basis on X iff B The entire neighborhood system , which is Nx (TB1) X = B. Thus each x X must be- recovered from the base by forming all supersets of B ∈T B long to some B which implies∈the existence the basic neighborhoods, is trivially a local base at T of a local baseSat ∈eachBpoint x X. x; non-trivial examples are given below. ∈ The second example of a base, consisting as (TB2) The intersection of any two members B1 and B of with x B B contains another mem- usual of a subcollection of a given collection, is the 2 TB ∈ 1 2 ber of T : (B1, B2 T ) (x B1 B2) ( B topological base T that allows the specification of B ∈ TB ∧ ∈ ⇒ ∃ ∈ B T : x B B1 B2). the topology on a set X in terms of a smaller col- B ∈ ⊆ T lection of open sets. This theorem, togetherT with Eq. (A.4) ensures that a given collection of subsets of a set X sat- Definition A.1.2. A base T in a topological space (X, ) is a subcollection of Bthe topology having isfying (TB1) and (TB2) induces some topology the propU erty that each U contains someU mem- on X; compared to this is the result that any ∈ U collection of subsets of a set X is a subbasis ber of T . Thus B for some topology on X. If X, however, already def = B : B U for each U (A.3) has a topology imposed on it, then Eq. (A.3) TB { ∈ U ⊆ ∈ U} U December 3, 2003 12:13 00851

Toward a Theory of Chaos 3209

must also be satisfied in order that the topol- R2. Of course, the entire neighborhood system at ogy generated by T is indeed . The next the- any point of a topological space is itself a (less use- B U orem connects the two types of bases of Defini- ful) local base at that point. By Theorem A.1.2, tions A.1.1 and A.1.2 by asserting that although B (x; d), D (x; d), ε > 0, B (x; d), Q q > 0 and ε ε q 3 a local base of a space need not consist of open B1/n(x; d), n Z+, for all x X are examples of sets and a topological base need not have any ref- bases in a metrizable∈ space with∈ topology induced erence to a point of X, any subcollection of the by a metric d. base containing a point is a local base at that point. In terms of local bases and bases, it is now possible to formulate the notions of first and second Theorem A.1.2. A collection of open sets T is countability as follows. a base for a topological space (X, ) iff for eBach x X, the subcollection U Definition A.1.3. A topological space is first ∈ countable if each x X has some countable neigh- = B : x B (A.5) ∈ Bx { ∈ U ∈ ∈ TB} borhood base, and is second countable if it has a of basic sets containing x is a local base at x. countable base. Every metrizable space (X, d) is first countable Proof. Necessity. Let be a base of (X, ) and T as both B(x, q) and B(x, 1/n) are N be a neighborhood ofBx, so that x U UN for Q3q>0 n∈Z+ examples{of coun}table neighb{orhood bases} at any some open set U = B and basic∈ op⊆en sets B ∈ TB x (X, d); hence Rn is first countable. It should be B. Hence x B N shows, from Eq. (A.1), that clear∈ that although every second countable space B is a lo∈cal ⊆basicS set at x. x is first countable, only a countable first countable ∈Sufficiency.B If U is an open set of X contain- space can be second countable, and a common ex- ing x, then the definition of local base Eq. (A.1) ample of an uncountable first countable space that requires x B U for some subcollection of basic x is also second countable is provided by Rn. Metriz- sets B in∈ ; hence⊆ U = B . By Eq. (A.4) x Bx x∈U x able spaces need not be second countable: any un- therefore, T is a topological base for X. B S countable set having the discrete topology is as an example. Because the basic sets are open, (TB2) of Theorem A.1.1 leads to the following physically Example A.1.2. The following is an important ex- appealing paraphrase of Theorem A.1.2. ample of a space that is not first countable as it is needed for our pointwise biconvergence of Sec. 3. Corollary. A collection of open sets of (X, ) TB U Let Map(X, Y ) be the set of all functions between is a topological base that generates iff for each the uncountable spaces (X, ) and (Y, ). Given open set U of X and each x U therUe is an open U V ∈ any integer I 1, and any finite collection of points set B T such that x B U; that is iff I ≥ I ∈ B ∈ ⊆ (xi)i=1 of X and of open sets (Vi)i=1 in Y , let x U ( B T : x B U) . B((x )I ; (V )I ) = g Map(X, Y ) : ∈ ∈ U ⇒ ∃ ∈ B ∈ ⊆ i i=1 i i=1 { ∈ Example A.1.1. Some examples of local bases in R (g(xi) Vi)(i = 1, 2, . . . , I) ∈ } are intervals of the type (x ε, x+ε), [x ε, x+ε] for (A.6) real ε, (x q, x+q) for rational− q, (x 1−/n, x+1/n) for n Z− , while for a metrizable −space with the be the functions in Map(X, Y ) whose graphs pass ∈ + I I topology induced by a metric d, each of the follow- through each of the sets (Vi)i=1 at (xi)i=1, and ing is a local base at x X: B (x; d) := y X : let T be the collection of all such subsets of ∈ ε { ∈ B I d(x, y) < ε and D (x; d) := y X : d(x, y) ε Map(X, Y ) for every choice of I, (xi)i=1, and ε I } { ∈ ≤ } (Vi) . The existence of a unique topology — the for ε > 0, Bq(x; d) for Q q > 0 and B1/n(x; d) i=1 T 2 3 topology of pointwise convergence on Map(X, Y ) — for n Z+. In R , two neighborhood bases at any x R∈2 are the disks centered at x and the set that is generated by the open sets B of the collec- ∈ tion now follows because of all squares at x with sides parallel to the axes. TB Although these bases have no elements in common, (TB1) is satisfied: For any f Map(X, Y ) there they are nevertheless equivalent in the sense that must be some x X and a corresp∈ onding V Y they both generate the same (usual) topology in such that f(x) ∈V , and ⊆ ∈ December 3, 2003 12:13 00851

3210 A. Sengupta

(TB2) is satisfied because the space (X, ) is not first countable (and as seen U I I J J above this is not a rare situation), it is not diffi- B((si)i=1; (Vi)i=1) B((tj)j=1; (Wj)j=1) cult to realize that sequences are inadequate to de- I J I J scribe convergence in X simply because it can have = B((si)i=1, (tj)\j=1; (Vi)i=1, (Wj)j=1) only countably many values whereas the space may implies that a function simultaneously belonging to require uncountably many neighborhoods to com- the two open sets on the left must pass through each pletely define the neighborhood system at a point. of the points defining the open set on the right. The resulting uncountable generalizations of a se- We now demonstrate that (Map(X, Y ), ) is T quence in the form of nets and filters is achieved not first countable by verifying that it is not through a corresponding generalization of the index possible to have a countable local base at any set N to the directed set D. f Map(X, Y ). If this is not indeed true, let I ∈ I I Definition A.1.4. A directed set D is a preordered Bf ((xi)i=1; (Vi)i=1) = g Map(X, Y ) : (g(xi) I { ∈ ∈ set for which the order , known as a direction of Vi) , which denotes those members of T that i=1} B D, satisfies contain f with Vi an open neighborhood of f(xi) in Y , be a countable local base at f, see Theo- (a) α D α α (that is is reflexive). rem A.1.2. Since X is uncountable, it is now pos- (b) α,∈β, γ⇒ Dsuch that (α β β γ) α γ sible to choose some x∗ X different from any of (that is∈ is transitive). ∧ ⇒ I ∈ ∗ the (xi)i=1 (for example, let x R be an irrational (c) α, β D γ D such that (α γ β γ). I ∈∗ ∗ ∗ ∈ ⇒ ∃ ∈ ∧ for rational (xi)i ), and let f(x ) V where V is an open neighborhood of f(x∗). ∈Then B(x∗; V ∗) While the first two properties are obvious enough, is an open set in Map(X, Y ) containing f; hence the third which replaces antisymmetry, ensures that from the definition of the local base, Eq. (A.1), or for any finite number of elements of the directed set, equivalently from the Corollary to Theorem A.1.2, there is always a successor (upper bound). Exam- there exists some (countable) I N such that ples of directed sets can be both straight forward, f BI B(x∗; V ∗). However, ∈ as any totally ordered set like N, R, Q, or Z and ∈ ⊆ all subsets of a set X under the superset or subset y V , if x = x , and 1 i I i ∈ i i ≤ ≤ relation (that is ( (X), ) or ( (X), ) that are f ∗(x) = y∗ V ∗, if x = x∗ directed by their Pusual ordering,⊇ Pand not⊆ quite so  6∈  arbitrary, otherwise obvious as the following examples which are signifi- cantly useful in dealing with convergence questions I is a simpleexample of a function on X that is in B in topological spaces, amply illustrate. (as it is immaterial as to what values the function The neighborhood system takes at points other than those defining BI), but ∗ ∗ DN = N : N not in B(x ; V ). From this it follows that a suffi- { ∈ Nx} cient condition for the topology of pointwise conver- at a point x X, directed by the reverse inclusion gence to be first countable is that X be countable. direction defined∈ as Even though it is not first countable, M N N M for M, N x , (A.7) ⇔ ⊆ ∈ N (Map(X, Y ), ) is a Hausdorff space when Y is T is a fundamental example of a natural direction of Hausdorff. Indeed, if f, g (Map(X, Y ), ) with x. In fact while reflexivity and transitivity are f = g, then f(x) = g(x)∈for some x TX. But N 6 6 ∈ clearly obvious, (c) follows because for any M, N then as Y is Hausdorff, it is possible to choose dis- , M M N and N M N. Of course, this∈ Nx joint open intervals Vf and Vg at f(x) and g(x) direction is not a total ordering on x. A more nat- respectively. urally useful Tdirected set in conTvergenceN theory is With this background on first and second DN = (N, t) : (N )(t N) (A.8) countability, it is now possible to go back to the t { ∈ Nx ∈ } question of nets, filters and sequences. Technically, under its natural direction a sequence on a set X is a map x : N X from the (M, s) (N, t) N M for M, N x ; set of natural numbers to X; instead→ of denoting ⇔ ⊆ ∈ N this is in the usual functional manner of x(i) with (A.9) i N, it is the standard practice to use the nota- DN is more useful than DN because, unlike the ∈ t tion (xi)i∈N for the terms of a sequence. However, if latter, DNt does not require a simultaneous choice December 3, 2003 12:13 00851

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of points from every N x that implicitly in- Deﬁnition A.1.7. A net χ : D X converges to volves a simultaneous application∈ N of the Axiom of x X if it is eventually in every→neighborhood of Choice; see Example A.1.3 below. The general in- x,∈that is dexed variation ( N )( µ D)(χ(ν µ) N) . ∀ ∈ Nx ∃ ∈ ∈ DN = (N, β) : (N )(β D)(x N) β { ∈ Nx ∈ β ∈ } The point x is known as the limit of χ and the col- (A.10) lection of all limits of a net is the limit set of Eq. (A.8), with natural direction lim(χ) = x X : ( N )( R Res(D)) { ∈ ∀ ∈ Nx ∃ β ∈ (M, α) (N, β) (α β) (N M) , (A.11) (χ(Rβ) N) (A.12) ≤ ⇔ ∧ ⊆ ⊆ } often proves useful in applications as will be clear of χ, with the set of residuals Res(D) in D given by from the proofs of Theorems A.1.3 and A.1.4. Res(D) = R (D) : R = β D { α ∈ P α { ∈ for all β α D . (A.13) Deﬁnition A.1.5 (Net). Let X be any set and D ∈ }} a directed set. A net χ : D X in X is a function The net adheres at x X 27 if it is frequently in → on the directed set D with values in X. every neighborhood of x∈, that is

A net, to be denoted as χ(α), α D, is there- (( N x)( µ D))(( ν µ) : χ(ν) N) . ∈ ∀ ∈ N ∀ ∈ ∃ ∈ fore a function indexed by a directed set. We adopt The point x is known as the adherent of χ and the the convention of denoting nets in the manner of collection of all adherents of χ is the adherent set functions and do not use the sequential notation χα of the net, which may be expressed in terms of the that can also be found in the literature. Thus, while cofinal subset of D every sequence is a special type of net, χ : Z X → Cof(D) = C (D) : C = β D is an example of a net that is not a sequence. { α ∈ P α { ∈ Convergence of sequences and nets are de- for some β α D (A.14) ∈ }} scribed most conveniently in terms of the notions of (thus Dα is cofinal in D iff it intersects every residual being eventually in and frequently in every neigh- in D), as borhood of points. We describe these concepts in adh(χ) = x X : ( N )( C Cof(D)) terms of nets which apply to sequences with obvi- { ∈ ∀ ∈ Nx ∃ β ∈ ous modifications. (χ(C ) N) . (A.15) β ⊆ } Definition A.1.6. A net χ : D X is said to be This recognizes, in keeping with the limit set, each → subnet of a net to be a net in its own right, and is (a) Eventually in a subset A of X if its tail is even- equivalent to tually in A: ( β D) : ( γ β)(χ(γ) A). ∃ ∈ ∀ ∈ adh(χ) = x X : ( N )( R Res(D)) (b) Frequently in a subset A of X if for any index { ∈ ∀ ∈ Nx ∀ α ∈ β D, there is a successor index γ D such (χ(Rα) N = ) . (A.16) that∈ χ(γ) is in A: ( β D)( γ β) ∈: (χ(γ) 6 ∅ } A). ∀ ∈ ∃ ∈ Intuitively, a sequence\ is eventually in a set A if it is always in it after a finite number of terms (of It is not difficult to appreciate that course, the concept of a finite number of terms is unavailable for nets; in this case the situation may (i) A net eventually in a subset is also frequently be described by saying that a net is eventually in A in it but not conversely, if its tail is in A) and it is frequently in A if it always (ii) A net eventually (respectively, frequently) in a returns to A to leave it again. It can be shown that subset cannot be frequently (respectively, even- a net is eventually (resp. frequently) in a set iff it is tually) in its complement. not frequently (resp. eventually) in its complement. With these notions of eventually in and fre- The following examples illustrate graphically quently in, convergence characteristics of a net may the role of a proper choice of the index set D in be expressed as follows. the description of convergence.

27This is also known as a cluster point; we shall, however, use this new term exclusively in the sense of the elements of a derived set, see Deﬁnition 2.3. December 3, 2003 12:13 00851

3212 A. Sengupta

Example A.1.3. (1) Let γ D. The eventually to yield a self-consistent tool for the description of constant net χ(δ) = x for δ ∈γ converges to x. convergence. As compared with sequences where, the index (2) Let x be a neighborhood system at a point x in N set is restricted to positive integers, the considerable X and suppose that the net (χ(N))N∈Nx is defined by freedom in the choice of directed sets as is abun- dantly borne out by the two preceding examples, def χ(M) = s M ; (A.17) is not without its associated drawbacks. Thus as ∈ a trade-off, the wide range of choice of the directed here the directed index set DN is ordered by the sets may imply that induction methods, so common natural direction (A.7) of x. Then χ(N) x be- N → in the analysis of sequences, need no longer apply cause given any x-neighborhood M DN, it follows from ∈ to arbitrary nets. (4) The non-convergent nets (actually these are M N DN χ(N) = t N M (A.18) ∈ ⇒ ∈ ⊆ sequences) that a point in any subset of M is also in M; χ(N) (a) (1, 1, 1, 1, . . .) adheres at 1 and 1 and is therefore eventually in every neighborhood of x. − n, − if n is odd − (b) x = (3) This slightly more general form of the previous n 1 1/(1 + n), if n is even example provides a link between the complimentary − concepts of nets and filters that is considered below. adheres at 1 for its even terms, but is unbounded For a point x X, and M, N x with the corre- in the odd terms. sponding directed∈ set M of Eq.∈ N(A.8) ordered by D s A converging sequence or net is also adhering its natural order (A.9), the net but, as examples (4) show, the converse is false. def χ(M, s) = s (A.19) Nevertheless it is true, as again is evident from examples (4), that in a first countable space where converges to x because, as in the previous example, sequences suffice, a sequence (xn) adheres to x iff for any given (M, s) DNs, it follows from ∈ some subsequence (xnm )m∈N of (xn) converges to x. (M, s) (N, t) DM χ(N, t) = t N M If the space is not first countable this has a corre- ∈ s ⇒ ∈ ⊆ (A.20) sponding equivalent formulation for nets with subnets replacing subsequences as follows. that χ(N, t) is eventually in every neighborhood Let (χ(α))α∈D be a net. A subnet of χ(α) is M of x. The significance of the directed set DNt the net ζ(β) = χ(σ(β)), β E, where σ : (E, ) of Eq. (A.8), as compared to DN, is evident from (D, ) is a function that captures∈ the essence of≤ the→ the net that it induces without using the Axiom of subsequential mapping n nm in N by satisfying Choice: For a subset A of X, the net χ(N, t) = t A 7→ indexed by the directed set ∈ (SN1) σ is an increasing order-preserving function: it respects the order of E: σ(β) σ(β0) for every DNt = (N, t) : (N x)(t N A) (A.21) β β0 E, and { ∈ N ∈ } ≤ ∈ under the direction of Eq. (A.9), convTerges to x X (SN2) For every α D there exists a β E such with all such x defining the closure Cl(A) of A. ∈Fur- that α σ(β). ∈ ∈ thermore taking the directed set to be These generalize the essential properties of a subse-

DNt = (N, t) : (N x)(t N A x ) quence in the sense that (1) Even though the index { ∈ N ∈ − { } } sets D and E may be different, it is necessary that (A.22) \ the values of E be contained in D, and (2) There which, unlike Eq. (A.21), excludes the point x that are arbitrarily large α D such that χ(α = σ(β)) may or may not be in the subset A of X, induces is a value of the subnet∈ ζ(β) for some β E. Re- the net χ(N, t) = t A x converging to x X, calling the first of the order relations Eq.∈(38) on with the set of all suc∈ h−x{yielding} the derived∈ set Map(X, Y ), we will denote a subnet ζ of χ by ζ χ. Der(A) of A. In contrast, Eq. (A.21) also includes We now consider the concept of filter on aset the isolated points t = x of A so as to generate X that is very useful in visualizing the behavior its closure. Observe how neighborhoods of a point, of sequences and nets, and in fact filters constitute which define convergence of nets and filters in a an alternate way of looking at convergence ques- topological space X, double up here as index sets tions in topological spaces. A filter on a set X F December 3, 2003 12:13 00851

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is a collection of nonempty subsets of X satisfying space and a filter on X. Then properties (F1) (F3) below that are simply those F − lim( ) = x X : ( N x)( F )(F N) of a neighborhood system x without specification F { ∈ ∀ ∈ N ∃ ∈ F ⊆ } of the reference point x. N (A.23) (F1) The empty set does not belong to , and ∅ F (F2) The intersection of any two members of a fil- adh( ) = x X : ( N x)( F ) ter is another member of the filter: F , F F { ∈ ∀ ∈ N ∀ ∈ F 1 2 ∈ (F N = ) (A.24) F1 F2 , 6 ∅ } F ⇒ ∈ F (F3) Every superset of a member of a filter belongs are respectively the\sets of limit points and adherent T to the filter: (F ) (F G) G ; in points of 28 F particular X ∈.F ∧ ⊆ ⇒ ∈ F ∈ F A comparison of Eqs. (A.12) and (A.16) with Example A.1.4 Eqs. (A.23) and (A.24) respectively demonstrate their formal similarity; this inter-relation between (1) The indiscrete filter is the smallest filter on X. filters and nets will be made precise in Defini- (2) The neighborhood system is the important Nx tions A.1.10 and A.1.11 below. It should be clear neighborhood filter at x on X, and any local from the preceding two equations that base at x is also a filter-base for x. In general for any subset A of X, N X N: A Int(N) lim( ) adh( ) , (A.26) { ⊆ ⊆ } F ⊆ F is a filter on X at A. with a similar result (3) All subsets of X containing a point x X is the principal filter (x) on X at x. More∈ gener- lim(χ) adh(χ) (A.27) FP ⊆ ally, if consists of all supersets of a nonempty holding for nets because of the duality between nets F subset A of X, then is the principal filter and filters as displayed by Definitions A.1.9 and F (A) = N X : A Int(N) at A. By A.1.10 below, with the equality in Eqs. (A.26) and FP { ⊆ ⊆ } adjoining the empty set to this filter give the (A.27) being true (but not characterizing) for ultra- p-inclusion and A-inclusion topologies on X, re- filters and ultranets respectively, see Example 4.2(3) spectively. The single element sets x and for an account of this notion. It should be clear from {{ }} A are particularly simple examples of filter- the equations of Definition A.1.8 that {bases} that generate the principal filters at x adh( ) = x X : ( a finer filter on X) and A. F { ∈ ∃ G ⊇ F ( x) (A.28) (4) For an uncountable (resp. infinite) set X, all G → } cocountable (resp. cofinite) subsets of X consti- consists of all the points of X to which some finer tute the cocountable (resp. cofinite or Frechet) filter (in the sense that implies every ele- filter on X. Again, adding to these filters the ment Gof is also in ) conFverges⊆ G in X; thus empty set gives the respective topologies. F G adh( ) = lim( : ) , Like the topological and local bases T and x F G G ⊇ F respectively, a subclass of may be usedBto defineB which corresponds to[ the net-result of Theo- F a filter-base F that in turn generate on X, just rem A.1.5 below, that a net χ adheres to x iff there as it is possibleB to define the conceptsFof limit and is some subnet of χ that converges to x in X. Thus adherence sets for a filter to parallel those for nets if ζ χ is a subnet of χ and is a filter coarser than then F ⊆ G that follow straightforwardly from Definition A.1.7, G taken with Definition A.1.11. lim(χ) lim(ζ) lim( ) lim( ) ⊆ F ⊆ G Definition A.1.8. Let (X, ) be a topological adh(ζ) adh(χ) adh( ) adh( ) ; T ⊆ G ⊆ F 28The restatement

x x (A.25) F → ⇔ N ⊆ F of Eq. (A.23) that follows from (F3), and sometimes taken as the definition of convergence of a filter, is significant as it ties up the algebraic filter with the topological neighborhood system to produce the filter theory of convergence in topological spaces. From the defining properties of it follows that for each x X, x is the coarsest (that is smallest) filter on X that F ∈ N converges to x. December 3, 2003 12:13 00851

3214 A. Sengupta

a filter finer than a given filter corresponds supersets . ( ) := is the smallest fil- G F F Σ∧ F F Σ∧ to a subnet ζ of a given net χ. The implication of ter on X thatS conFtainsS andS is the filter generated FS this correspondence should be clear from the asso- by F . ciation between nets and filters contained in Defini- EquationS (A.24) can be put in the more useful tions A.1.10 and A.1.11. and transparent form given by A filter-base in X is a non-empty family Theorem A.1.3. For a filter in a space (X, ) (B ) D = of subsets of X characterized by α α∈ FB F T adh( ) = Cl(F ) (FB1) There are no empty sets in the collection F : F ( α D)(B = ) B F ∈F ∀ ∈ α 6 ∅ \ (A.31) (FB2) The intersection of any two members of = Cl(B) , FB contains another member of : B , B B ∈ FB FB α β ∈ FB ⇒ \ ( B : B B B ); and dually adh(χ), are closed sets. ∃ ∈ FB ⊆ α β hence any class of subsets of X that does not con- T Proof. Follows immediately from the definitions for tain the empty set and is closed under finite inter- the closure of a set Eq. (20) and the adherence of a sections is a base for a unique filter on X; compare filter Eq. (A.24). As always, it is a matter of conve- the properties (NB1) and (NB2) of a local basis nience in using the basic filters instead of to given at the beginning of this Appendix. Similar to F generate the adherence set. B F Definition A.1.1 for the local base, it is possible to define It is in fact true that the limit sets lim( ) and lim(χ) are also closed set of X; the argumenFts in- Definition A.1.9. A filter-base in a set X is a F volving ultrafilters are omitted. subcollection of the filter on X Bhaving the prop- Similar to the notion of the adherence set of erty that each F containsF some member of . F a filter is its core — a concept that unlike the ad- Thus ∈ F B herence, is purely set-theoretic being the infimum def F = B : B F for each F (A.29) of the filter and is not linked with any topological B { ∈ F ⊆ ∈ F} structure of the underlying (infinite) set X — de- determines the filter fined as

= F X : B F for some B F (A.30) core( ) = F . (A.32) F { ⊆ ⊆ ∈ B} F F ∈F reciprocally as all supersets of the basic elements. \ From Theorem A.1.3 and the fact that the closure This is the smallest filter on X that contains and of a set A is the smallest closed set that contains A, FB is said to be the filter generated by its filter-base F ; see Eq. (25) at the end of Tutorial 4, it is clear that alternatively is the filter-base of . The entireB in terms of filters FB F neighborhood system x, the local base x, x A A = core(F (A)) for x Cl(A), and theN set of all residualsB ofN a di- P Cl(A) = adh(F (A)) (A.33) rected∈set D are among the most useful examplesTof P = core(Cl( (A))) filter-bases on X, A and D respectively. Of course, FP every filter is trivially a filter-base of itself, and the where F (A) is the principal filter at A; thus the singletons x , A are filter-bases that generate core andPadherence sets of the principal filter at A {{ }} { } the principal filters F (x) and F (A) at x, and A are equal respectively to A and Cl(A) — a classic ex- respectively. P P ample of equality in the general relation Cl( A ) α ⊆ Paralleling the case of topological subbase T , Cl(Aα) — but both are empty, for example, in S T a filter subbase F can be defined on X to be any the case of an infinitely decreasing family of ratio- collection of subsetsS of X with the finite intersection Tnals centered at any irrational (leading to a princi- property (as compared with T where no such condi- pal filter-base of rationals at the chosen irrational). tion was necessary, this represenS ts the fundamental This is an important example demonstrating that point of departure between topology and filter) and the infinite intersection of a non-empty family of it is not difficult to deduce that the filter generated (closed ) sets with the finite intersection property by F on X is obtained by taking all finite inter- may be empty, a situation that cannot arise on sectionsS of members of followed by their a finite set or an infinite compact set. Filters on FS∧ FS December 3, 2003 12:13 00851

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X with an empty core are said to be free, and (ii) χ is frequently in A ( R Res(D)) ⇒ ∀ α ∈ are fixed otherwise: notice that by its very defini- (A χ(Rα) = ) A χ = . tion filters cannot be free on a finite set, and a 6 ∅ ⇒ F 6 ∅ free filter represents an additional feature that may Limits Tand adherences areTobviously preserved in arise in passing from finite to infinite sets. Clearly switching between nets (respectively, filters) and (adh( ) = ) (core( ) = ), but as the im- the filters (respectively, nets) that they generate: F ∅ ⇒ F ∅ portant example of the rational space in the reals lim(χ) = lim( χ), adh(χ) = adh( χ) (A.34) illustrate, the converse need not be true. Another F F example of a free filter of the same type is provided lim( ) = lim(χF ), adh( ) = adh(χF ) . (A.35) F F by the filter-base [a, ) : a R in R. Both these The proofs of the two parts of Eq. (A.34), for ex- examples illustrate{ the∞importan∈ }t property that a ample, go respectively as follows. x lim(χ) χ is filter is free iff it contains the cofinite filter, and the eventually in ( N )( F ∈ ) suc⇔h that cofinite filter is the smallest possible free filter on an x x χ (F N) xN ⇔lim∀( ),∈ Nand ∃x ∈adh(F χ) χ is infinite set. The free cofinite filter, as these examples χ frequen⊆ tly⇔in ∈ ( FN )( F∈ )(N⇔ F = illustrate, may be typically generated as follows. Let x x χ ) x adhN( ⇔); here∀ F∈ Nis a sup∀ erset∈ F of χ(R ).6 A be a subset of X, x Bdy (A), and consider χ α X−A ∅ ⇔Some∈ examplesF of convergence of filters areT the directed set Eq. (A.21)∈ to generate the corresponding net in A given by χ(N , t) = t A. (1) Any filter on an indiscrete space X converges ∈ Nx ∈ Quite clearly, the core of any Frechet filter based on to every point of X. this net must be empty as the point x does not lie (2) Any filter on a space that coincides with its in A. In general, the intersection is empty because topology (minus the empty set, of course) con- if it were not so then the complement of the inter- verges to every point of the space. section — which is an element of the filter — would (3) For each x X, the neighborhood filter x be infinite in contravention of the hypothesis that converges to∈x; this is the smallest filter onNX the filter is Frechet. It should be clear that every fil- that converges to x. ter finer than a free filter is also free, and any filter (4) The indiscrete filter = X converges to no coarser than a fixed filter is fixed. point in the space (XF, ,{A,}X A, X ), but Nets and filters are complimentary concepts converges to every poin{∅t of X A− if X has} the and one may switch from one to the other as fol- topology , A, X because the− only neighbor- lows. hood of an{∅y point }in X A is X which is contained in the filter. − Definition A.1.10. Let be a filter on X and let F DFx = (F, x) : (F )(x F ) be a directed set One of the most significant consequences of con- with its{ natural direction∈ F (F∈, x)} (G, y) (G vergence theory of sequences and nets, as shown by ⇒ ⊆ F ). The net χF : DFx X defined by the two theorems and the corollary following, is that → this can be used to describe the topology of a set. χ (F, x) = x F The proofs of the theorems also illustrate the close is said to be associated with the filter , see inter-relationship between nets and filters. Eq. (A.20). F Theorem A.1.4. For a subset A of a topological Definition A.1.11. Let χ : D X be a net and space X, R = β D : β α D a residual→ in D. Then α { ∈ ∈ } Cl(A) = x X : ( a net χ in A)(χ x) . def { ∈ ∃ → } = χ(R ) : Res(D) X for all α D (A.36) FBχ { α → ∈ } is the filter-base associated with χ, and the corre- Proof. Necessity. For x Cl(A), construct a net sponding filter χ obtained by taking all supersets ∈ χ x in A as follows. Let x be a topological local of the elements ofF is the filter associated with χ. → B FBχ base at x, which by definition is the collection of all open sets of X containing x. For each β D, the F χ is a filter-base in X because χ( Rα) B ⊆ sets ∈ χ(Rα), that holds for any functional relation, T proves (FB2). It is not difficult to verify that N = B : B T β { α α ∈ Bx} (i) χ is eventually in A A , and αβ ⇒ ∈ Fχ \ December 3, 2003 12:13 00851

3216 A. Sengupta

form a nested decreasing local neighborhood fil- Theorem A.1.5. If χ is a net in a topological space ter base at x. With respect to the directed set X, then x adh(χ) iff some subnet ζ(β) = χ(σ(β)) ∈ DNβ = (Nβ, β) : (β D)(xβ Nβ) of Eq. (A.10), of χ(α), with α D and β E, converges in X to define the{ desired net∈in A by∈ } x; thus ∈ ∈ adh(χ)= x X : ( a subnet ζ χ in X)(ζ x) . χ(Nβ, β) = xβ Nβ A { ∈ ∃ → } ∈ (A.39) where the family of non-empty decreasing\ subsets N A of X constitute the filter-base in A as re- Proof. Necessity. Let x adh(χ). Define a subnet β ∈ quired by the directed set N . It now follows from function σ :D Nα D by σ(Nα, α) = α where DNα D β → Eq.T(A.11) and the arguments in Example A.1.3(3) is the directed set of Eq. (A.10): (SN1) and (SN2) that xβ x; compare the directed set of Eq. (A.21) are quite evidently satisfied according to Eq. (A.11). for a more→ compact, yet essentially identical, argu- Proceeding as in the proof of the preceding theorem ment. Carefully observe the dual roles of as a it follows that xβ = χ(σ(Nα, α)) = ζ(Nα, α) x x → neighborhood filter base at x. N is the required converging subnet that exists from Sufficiency. Let χ be a net in A that con- Eq. (A.15) and the fact that χ(Rα) Nα = for 6 ∅ verges to x X. For any N , there is a every Nα x, by hypothesis. α x ∈ N T R Res(D)∈of Eq. (A.13) such that∈ Nχ(R ) N . Sufficiency. Assume now that χ has a subnet α ∈ α ⊆ α Hence the point χ(α) = xα of A belongs to Nα so ζ(Nα, α) that converges to x. If χ does not adhere that A N = which means, from Eq. (20), that at x, there is a neighborhood Nα of x not frequented α 6 ∅ x Cl(A). by it, in which case χ must be eventually in X Nα. ∈ T Then ζ(N , α) is also eventually in X N so−that α − α Corollary. Together with Eqs. (20) and (22), it fol- ζ cannot be eventually in Nα, a contradiction of the lows that hypothesis that ζ(N , α) x.29 α → Der(A) = x X : ( a net ζ in A x )(ζ x) Equations (A.36) and (A.39) imply that the clo- { ∈ ∃ − { } → } (A.37) sure of a subset A of X is the class of X-adherences of all the (sub)nets of X that are eventually in A. The filter forms of Eqs. (A.36) and (A.37) This includes both the constant nets yielding the Cl(A) = x X : ( a filter on X) isolated points of A and the non-constant nets lead- {(A ∈ )( ∃ x) F ing to the cluster points of A, and implies the fol- ∈ F F → } (A.38) lowing physically useful relationship between con- Der(A) = x X : ( a filter on X) vergence and topology that can be used as defining {(A ∈ x ∃ )( Fx) − { } ∈ F F → } criteria for open and closed sets having a more appealing physical significance than the original def- then follows from Eq. (A.25) and the finite inter- initions of these terms. Clearly, the term “net” is section property (F2) of so that every neighborhood of x must intersect AF(respectively A x ) in justifiably used here to include the subnets too. Eq. (A.38) to produce the converging net −ne{ede}d in The following corollary of Theorem A.1.5 sum- the proof of Theorem A.1.3. marizes the basic topological properties of sets in terms of nets (respectively, filters). We end this discussion of convergence in topo- Corollary. Let A be a subset of a topological space logical spaces with a proof of the following theorem X. Then which demonstrates the relationship that “eventually in” and “frequently in” bears with each other; (1) A is closed in X iff every convergent net of Eq. (A.39) below is the net-counterpart of the filter X that is eventually in A actually converges equation (A.28). to a point in A (respectively, iff the adhering

29In a ﬁrst countable space, while the corresponding proof of the ﬁrst part of the theorem for sequences is essentially the same as in the present case, the more direct proof of the converse illustrates how the convenience of nets and directed sets may

require more general arguments. Thus if a sequence (xi)i∈N has a subsequence (xik )k∈N converging to x, then a more direct line of reasoning proceeds as follows. Since the subsequence converges to x, its tail (xik )k≥j must be in every neighborhood N of x. But as the number of such terms is infinite whereas i : k < j is only finite, it is necessary that for any given n N, { k } ∈ cofinitely many elements of the sequence (xik )ik n be in N. Hence x adh((xi)i N). ≥ ∈ ∈ December 3, 2003 12:13 00851

Toward a Theory of Chaos 3217

points of each filter-base on A all belong to A). to x unless it is of the uncountable type30 Thus no X-convergent net in a closed subset may converge to a point outside it. (x0, x1, . . . , xI , xI+1, xI+1, . . .) (A.40) (2) A is open in X iff every convergent net of X with only a finite number I of distinct terms ac- that converges to a point in A is eventually in tually belonging to the closed sequential set F = A. Thus no X-convergent net outside an open X G, and xI+1 = x. Note that as we are concerned subset may converge to a point in the set. only− with the eventual behavior of the sequence, we (3) A is closed-and-open (clopen) in X iff every may discard all distinct terms from G by consider- convergent net of X that converges in A is even- ing them to be in F , and retain only the constant tually in A and conversely. sequence (x, x, . . .) in G. In comparison with the (4) x Der(A) iff some net (respectively, filter- ∈ cofinite case that was considered in Sec. 4, the en- base) in A x converges to x; this clearly tire countably infinite sequence can now lie outside eliminates the− {isolate} d points of A and x ∈ a neighborhood of x thereby enforcing the eventual Cl(A) iff some net (respectively, filter-base) in constancy of the sequence. This leads to a gener- A converges to x. alization of our earlier cofinite result in the sense that a cocountable filter on a cocountable space con- Remark. The differences in these characterizations verges to every point in the space. should be fully appreciated: If we consider the clus- It is now straightforward to verify that for a ter points Der(A) of a net χ in A as the resource point x in an uncountable cocountable space X generated by χ, then a closed subset of X can be 0 considered to be selfish as it keeps all it resource (a) Even though no sequence in the open set G = to itself: Der(A) A = Der(A). The opposite of X x can converge to x , yet x Cl(G) ∩ 0 0 0 this is a donor set that donates all its generated re- since− {the}intersection of any (uncountable)∈ open sources to its neighbor: Der(A) X A = Der(A), neighborhood U of x with G, being an un- ∩ − 0 while for a neutral set, both Der(A) A = and countable set, is not empty. ∩ 6 ∅ Der(A) X A = implying that the convergence (b) By Corollary 1 of Theorem A.1.5, the uncount- ∩ − 6 ∅ resources generated in A and X A can be de- able open set G = X x is also closed in X − − { 0} posited only in the respective sets. The clopen sets because if any sequence (x1, x2, . . .) in G con- (see diagram 2–2 of Fig. 22) are of some special verges to some x X, then x must be in G interest as they are boundary less so that no net- as the sequence must∈ be eventually constant in resources can be generated in this case as any such order for it to converge. But this is a contra- limit are required to be simultaneously in the set diction as G cannot be closed since it is not and its complement. countable.31 By the same reckoning, although x0 is not an open set because its complement Example A.1.2. (Continued). This continuation is{ not} countable, nevertheless it follows from Example A.1.2 illustrates how sequential conver- Eq. (A.40) that should any sequence converge gence is inadequate in spaces that are not first to the only point x0 of this set, then it must countable like the uncountable set with cocountable eventually be in x0 so by Corollary 2 of the topology. In this topology, a sequence can converge same theorem, x{ }becomes an open set. { 0} to a point x in the space iff it has only a finite num- (c) The identity map 1 : X Xd, where Xd is ber of distinct terms, and is therefore eventually X with discrete topology, is→not continuous be- constant. Indeed, let the complement cause the inverse image of any singleton of Xd is def not open in X. Yet if a sequence converges in X G = X F, F = x : x = x, i N − { i i 6 ∈ } to x, then its image (1(x)) = (x) must actually of the countably closed sequential set F be an open converge to x in Xd because a sequence con- neighborhood of x X. Because a sequence (xi)i∈N verges in a discrete space, as in the cofinite or in X converges to∈a point x X iff it is eventu- cocountable spaces, iff it is eventually constant; ally in every neighborhood (including∈ G) of x, the this is so because each element of a discrete sequence represented by the set F cannot converge space being clopen is boundary-less.

30This is uncountable because interchanging any two eventual terms of the sequence does not alter the sequence. 31Note that x is a 1-point set but (x) is an uncountable sequence. { } December 3, 2003 12:13 00851

3218 A. Sengupta

This pathological behavior of sequences in a dropping all basic open sets that do not inter- non Hausdorff, non first countable space does not sect. Then a (coarser) topology can be gener- arise if the discrete indexing set of sequences is re- ated from this base by taking all unions, and placed by a continuous, uncountable directed set a filter by taking all supersets according to like R for example, leading to nets in place of se- Eq. (A.30). For any given filter this expression quences. In this case the net can be in an open set may be used to extract a subclass F as a base without having to be constant valued in order to for . B converge to a point in it as the open set can be de- F fined as the complement of a closed countable part A.2. Initial and Final Topology of the uncountable net. The careful reader could The commutative diagram of Fig. contains four not have failed to notice that the burden of the sub-diagrams X X f(X), Y X f(X), above arguments, as also of that in the example B B X X Y and −X f(−X) Y . Of−these,−the first following Theorem 4.6, is to formalize the fact that B two−are esp− ecially significan− t−as they can be used to since a closed set is already defined as a countable conveniently define the topologies on X and f(X) (respectively finite) set, the closure operation cannot B from those of X and Y , so that f , f −1 and G add further points to it from its complement, and B B have some desirable continuity properties; we recall any sequence that converges in an open set in these that a function f : X Y is continuous if inverse topologies must necessarily be eventually constant images of open sets of→Y are open in X. This sim- at its point of convergence, a restriction that no ple notion of continuity needs refinement in order longer applies to a net. The cocountable topology that topologies on X and f(X) be unambiguously thus has the very interesting property of filtering B defined from those of X and Y , a requirement that out a countable part from an uncountable set, as leads to the concepts of the so-called final and initial for example the rationals in R. topologies. To appreciate the significance of these new constructs, note that if f: (X, ) (Y, ) is a This example serves to illustrate the hard truth continuous function, there may beUop→en setsV in X that in a space that is not first countable, the sim- that are not inverse images of open — or for that plicity of sequences is not enough to describe its matter of any — subset of Y , just as it is possible topological character, and in fact “sequential con- for non-open subsets of Y to contribute to . When vergence will be able to describe only those topolo- the triple , f, are tuned in such aUmanner gies in which the number of (basic) neighborhoods that these {Uare impVossible,} the topologies so gener- around each point is no greater than the number ated on X and Y are the initial and final topologies of terms in the sequences”, [Willard, 1970]. It is respectively; they are the smallest (coarsest) and important to appreciate the significance of this in- largest (finest) topologies on X and Y that make terplay of convergence of sequences and nets (and f : X Y continuous. It should be clear that every of continuity of functions of Appendix A.1) and the image→and preimage continuous function is contin- topology of the underlying spaces. uous, but the converse is not true. A comparison of the defining properties (T1)– Let sat(U) := f −f(U) X be the saturation (T3) of topology with (F1)–(F3) of that of the of an open set U of X and comp⊆ (V ) := ff −(V ) = filter , shows thatT a filter is very close to a topol- V f(X) Y be the component of an open set V ogy withF the main difference being with regard to of Y on the∈ range f(X) of f. Let , de- the empty set which must always be in but never sat comp noteT respectively the saturations UU = Vsat(U) : in . Addition of the empty set to a Tfilter yields sat U of the open sets of X and the comp{ onents a topF ology, but removal of the empty set from a V ∈ U=} comp(V ) : V of the open sets of Y topology need not produce the corresponding fil- comp whenever{these are also∈opVen} in X and Y respec- ter as the topology may contain nonintersecting tively. Plainly, and . sets. Usat ⊆ U Vcomp ⊆ V The distinction between the topological and Definition A.2.1. For a function e : X (Y, ), filter-bases should be carefully noted. Thus the preimage or initial topology of X →based Von (generated by) e and is V (a) While the topological base may contain the def IT e; = U X : U = e−(V ) if V , empty set, a filter-base cannot. { V} { ⊆ ∈ Vcomp} (b) From a given topology, form a common base by (A.41) December 3, 2003 12:13 00851

Toward a Theory of Chaos 3219

while for q : (X, ) Y , the image or final topol- (a) f is continuous iff g is continuous, ogy of Y based onU (generated→ by) and q is (b) f is preimage continuous iff = IT g; . U U1 { V} def FT ; q = V Y : q−(V ) = U if U . {U } { ⊆ ∈ Usat} As we need the second part of these theorems in (A.42) our applications, their proofs are indicated below. The special significance of the first parts is that they Thus, the topology of (X, IT e; ) consists of, and ensure the converse of the usual result that the com- only of, the e-saturations of{ allV}the open sets of position of two continuous functions is continuous, e(X), while the open sets of (Y, FT ; q ) are the namely that one of the components of a composition q-images in Y (and not just in q(X{U)) of }all the q- is continuous whenever the composition is so. saturated open sets of X.32 The need for defining (A.41) in terms of comp rather than will be- Proof of Theorem A.2.1. If f be image continuous, come clear in the folloV wing. The subspaceV topol- = V Y : f −(V ) and = U X : ogy IT i; of a subset A (X, ) is a basic ex- 1 1 1 1 1 1 1 1 Vq−(U {) ⊆ are the final∈ Utop} ologiesU of {Y and⊆ X ample {of theU} initial topology⊆by theU inclusion map 1 1 1 based on∈ theU} topologies of X and X, respectively. i : X A (X, ), and we take its generalization 1 Then = V Y : q−f −(V ) shows that h e : (A,⊇IT →e; ) U (Y, ) that embeds a subset A 1 1 1 1 is imageV con{tinuous.⊆ ∈ U} of X into {Y asV}the→prototVype of a preimage continu- Conversely, when h is image continuous, = ous map. Clearly the topology of Y may also contain 1 V Y : h−(V ) = V VY : open sets not in e(X), and any subset in Y e(X) 1 1 1 1 1 {q−f −⊆(V ) , with} ∈= UU} X{ : q⊆−(U ) may be added to the topology of Y without−alter- 1 1 1 1 1 , proves} f∈−U(V} ) to beUopen{in X⊆ and thereby ∈f ing the preimage topology of X: open sets of Y not 1 1 toU}be image continuous. in e(X) may be neglected in obtaining the preimage topology as e−(Y e(X)) = . The final topology on − ∅ Proof of Theorem A.2.2. If f be preimage a quotient set by the quotient map Q : (X, ) − X/ , which is just the collection of Q-images Uof the→ continuous, 1 = V1 Y1 : V1 = e (V ) if V ∼ and = VU {X ⊆: U = f −(V ) if V ∈ V} Q-saturated open sets of X, known as the quotient U1 { 1 ⊆ 1 1 1 1 ∈ V1} topology of X/ , is the basic example of the image are the initial topologies of Y1 and X1 respectively. ∼ Hence from = U X : U = f −e−(V ) if V topology and the resulting space (X/ , FT ; Q ) U1 { 1 ⊆ 1 1 ∈ is called the quotient space. We take the∼ generaliza-{U } it follows that g is preimage continuous. V} Conversely, when g is preimage continuous, tion q : (X, ) (Y, FT ; q ) of Q as the proto- − U → {U } 1 = U1 X1 : U1 = g (V ) if V = type of an image continuous function. U { ⊆ − − ∈ V} The following results are specifically useful in U1 X1 : U1 = f e (V ) if V and { =⊆ V Y : V = e−(V ) if V ∈ V}show dealing with initial and final topologies; compare V1 { 1 ⊆ 1 1 ∈ V} the corresponding results for open maps given later. that f is preimage continuous.

Theorem A.2.1. Let (X, ) and (Y1, 1) be topo- Since both Eqs. (A.41) and (A.42) are in terms U V logical spaces and let X1 be a set. If f : X1 of inverse images (the first of which constitutes a → (Y1, 1), q : (X, ) X1, and h = f q : direct, and the second an inverse, problem) the im- V U → ◦ (X, ) (Y1, 1) are functions with the topology age f(U) = comp(V ) for V is of interest as U → V ∈ V 1 of X1 given by FT ; q , then it indicates the relationship of the openness of f U {U } (a) f is continuous iff h is continuous, with its continuity. This, and other related concepts (b) f is image continuous iff = FT ; h . are examined below, where the range space f(X) is V1 {U } always taken to be a subspace of Y . Openness of Theorem A.2.2. Let (Y, ) and (X , ) be topo- a function f: (X, ) (Y, ) is the “inverse” of V 1 U1 U → V logical spaces and let Y1 be a set. If f: (X1, 1) continuity, when images of open sets of X are re- Y , e : Y (Y, ) and g = e f : (X , U ) → quired to be open in Y ; such a function is said to be 1 1 → V ◦ 1 U1 → (Y, ) are function with the topology 1 of Y1 given open. Following are two of the important properties by ITV e; , then V of open functions. { V} 32We adopt the convention of denoting arbitrary preimage and image continuous functions by e and q respectively even though they are not injective or surjective; recall that the embedding e : X A Y and the association q : X f(X) are 1 : 1 and ⊇ → → onto respectively. December 3, 2003 12:13 00851

3220 A. Sengupta

(1) If f : (X, ) (Y, f( )) is an open function, continuous. Indeed, from its injectivity and conti- U → U then so is f< : (X, ) (f(X), IT i; f( ) ). nuity, inverse images of all open subsets of Y are The converse is trueUif f→(X) is an open{ set Uof }Y ; saturated-open in X, and openness of f ensures that thus openness of f< : (X, ) (f(X), f<( )) these are the only open sets of X the condition of implies that of f : (X, )U →(Y, ) wheneverU injectivity being required to exclude non-saturated U → V f(X) is open in Y such that f<(U) for U sets from the preimage topology. It is therefore pos- . The truth of this last assertion ∈folVlows eas-∈ sible to rewrite Eq. (A.41) as U ily from the fact that if f<(U) is an open set of f(X) Y, then necessarily f<(U) = V f(X) U IT e; e(U) = V if V comp , (A.43) for some⊂ V , and the intersection of two ∈ { V} ⇔ ∈ V ∈ V T open sets of Y is again an open set of Y . and to compare it with the following criterion for an (2) If f: (X, ) (Y, ) and g : (Y, ) (Z, ) U → V V → W injective, open-continuous map f: (X, ) (Y, ) are open functions then g f: (X, ) (Z, ) that necessarily satisfies sat(A) = A forUall→A XV is also open. It follows that◦ theUcondition→ Win ⊆ (1) on f(X) can be replaced by the require- U ( f(U) = ) (f −1(V ) ). ment that the inclusion i : (f(X), IT i; ) ∈ U ⇔ {{ }U∈U Vcomp ∧ |V ∈V ∈U (Y, ) be an open map. This interc{hangeV} →of (A.44) f(XV) with its inclusion i: f(X) Y into Y is a basic result that finds application→ in many Final Topology. Since it is necessarily produced situations. on the range (q) of q, the final topology is often considered in Rterms of a surjection. This however Collected below are some useful properties of is not necessary as, much in the spirit of the ini- the initial and final topologies that we need in this tial topology, Y q(X) = inherits the discrete work. topology without−altering6 an∅ything, thereby allow- ing condition (A.42) to be restated in the following Initial Topology. In Fig. 21(b), consider Y1 = more transparent form h(X1), e i and f h< : X1 (h(X1), → − → − → IT i; ). From h (B) = h (B h(X1)) for any V FT ; q V = q(U) if U sat , (A.45) B { VY}, it follows that for an open set V of Y , ∈ {U } ⇔ ∈ U − ⊆ − T h (Vcomp) = h (V ) is an open set of X1 which, if and to compare it with the following criterion for the topology of X is IT h; , are the only open 1 { V} a surjective, open-continuous map f: (X, ) sets of X1. Because Vcomp is an open set of h(X1) in U → (Y, ) that necessarily satisfies f B = B for all its subspace topology, this implies that the preim- B VY age topologies IT h; and IT h ; IT i; of ⊆ { V} { < { V}} X1 generated by h and h< are the same. Thus the − V ( sat = f (V ) V ∈V ) (f(U) U ∈ U ). preimage topology of X1 is not affected if Y is re- ∈ V ⇔ U { } ∧ | ∈ V placed by the subspace h(X1), the part Y h(X1) (A.46) contributing nothing to IT h; . − A preimage continuous{functionV} e : X (Y, ) As may be anticipated from Fig. 21, the final topol- is not necessarily an open function. Indeed,→ if UV= ogy does not behave as well for subspaces as the ini- e−(V ) IT e; , it is almost trivial to verify tial topology does. This is so because in Fig. 21(a) along the∈ lines{ ofV}the restriction of open maps to the two image continuous functions h and q are its range, that e(U) = ee−(V ) = e(X) V , V , connected by a preimage continuous inclusion f, is open in Y (implying that e is an open map)∈ Viff whereas in Fig. 21(b) all the three functions are e(X) is an open subset of Y (becauseT finite in- preimage continuous. Thus quite like open func- tersections of open sets are open). A special case tions, although image continuity of h : (X, ) U → of this is the important consequence that the re- (Y1, FT ; h ) implies that of h< : (X, ) striction e : (X, IT e; ) (e(X), IT i; ) of (h(X), IT{U i; }FT ; h )) for a subspace h(UX) →of < { V} → { V} { {U } e : (X, IT h; ) (Y, ) to its range is an open Y1, the converse need not be true unless — en- map. Even{thoughV} →a preimageV continuous map need tirely like open functions again — either h(X) is not be open, it is true that an injective, continu- an open set of Y1 or i : (h(X), IT i; FT ; h )) ous and open map f : X (Y, ) is preimage (X, FT ; h ) is an open map.{ Since{Uan }op→en → V {U } December 3, 2003 12:13 00851

Toward a Theory of Chaos 3221

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(a) (b)

Fig. 21. Continuity in ﬁnal and initial topologies.

preimage continuous map is image continuous, this The following is a slightly more general form of makes i : h(X) Y1 an ininal function and hence the restriction on the inclusion that is needed for all the three legs→of the commutative diagram image image continuity to behave well for subspaces of Y . continuous. Theorem A.2.3. Let q : (X, ) (Y, FT ; q ) Like preimage continuity, an image continuous U → {U } function q : (X, ) Y need not be open. How- be an image continuous function. For a subspace B U → of (Y, FT ; q ), ever, although the restriction of an image continu- {U } ous function to the saturated open sets of its domain FT IT j; ; q< = IT i; FT ; q is an open function, q is unrestrictedly open iff the { { U} } { {U }} − saturation of every open set of X is also open in X. where q< : (q (B), IT j; ) (B, FT IT j; ; q ), if either q is an op{ enUmap} →or B is an{ op{enUset} In fact it can be verified without much effort that a <} continuous, open surjection is image continuous. of Y . Combining Eqs. (A.43) and (A.45) gives the fol- In summary we have the useful result that an lowing criterion for ininality open preimage continuous function is image continuous and an open image continuous function is U and V IFT sat; f; ∈ {U V} preimage continuous, where the second assertion ( f(U) = )( = f −(V ) ) , ⇔ { }U∈Usat V Usat { }V ∈V follows on neglecting non-saturated open sets in X; (A.47) this is permitted in as far as the generation of the final topology is concerned, as these sets produce which reduces to the following for a homeomor- the same images as their saturations. Hence an im- phism f that satisfies both sat(A) = A for A X ⊆ age continuous function q : X Y is preimage and f B = B for B Y → ⊆ continuous iff every open set in X is saturated with U and V HOM ; f; respect to q, and a preimage continuous function ∈ −1 {U V} e : X Y is image continuous iff the e-image of ( = f (V ) V ∈V )( f(U) U∈U = ) ⇔ U { } { } V every op→en set of X is open in Y . (A.48)

and compares with A.3. More on Topological Spaces U and V OC ; f; This Appendix — which completes the review ∈ {U V} of those concepts of topological spaces begun in (sat(U) : f(U) = ) ⇔ ∈ U { }U∈U Vcomp Tutorial 4 that are needed for a proper understand- (comp(V ) : f −(V ) = ) ing of this work — begins with the following sum- ∧ ∈ V { }V ∈V Usat (A.49) mary of the diﬀerent possibilities in the distribution of Der(A) and Bdy(A) between sets A X for an open-continuous f. and its complement X A, and follows it up with⊆ a − December 3, 2003 12:13 00851

3222 A. Sengupta

few other important topological concepts that have Lemma A.3.2. If A is a subspace of X, a sepa- been used, explicitly or otherwise, in this paper. ration of A is a pair of disjoint nonempty subsets H1 and H2 of A whose union is A neither of which Definition A.3.1 (Separation, Connected Space). contains a cluster point of the other. A is connected A separation (disconnection) of X is a pair of mutu- iff there is no separation of A. ally disjoint nonempty open (and therefore closed) subsets H and H such that X = H H . A space 1 2 1 ∪ 2 Proof. Let H1 and H2 be a separation of A so X is said to be connected if it has no separation, that they are clopen subsets of A whose union is that is, if it cannot be partitioned into two open or A. As H1 is a closed subset of A it follows that two closed non-empty subsets. X is separated (dis- H1 = ClX (H1) A, where ClX (H1) A is the clo- connected) if it is not connected. sure of H1 in A; hence ClX (H1) H2 = . But as the closure of aTsubset is the union ofTthe set∅ and its It follows from the definition, that for a dis- T adherents, an empty intersection signifies that H2 connected space X the following are equivalent cannot contain any of the cluster points of H1. A statements. similar argument shows that H1 does not contain (a) There exist a pair of disjoint non-empty open any adherent of H2. subsets of X that cover X, Conversely suppose that neither H1 nor H2 contain an adherent of the other: Cl (H ) H = (b) There exist a pair of disjoint non-empty closed X 1 2 ∅ subsets of X that cover X, and ClX (H2) H1 = . Hence ClX (H1) A = H1 ∅ T (c) There exist a pair of disjoint non-empty clopen and ClX (H2) A = H2 so that both H1 and H2 T T subsets of X that cover X, are closed in A. But since H1 = A H2 and H = A H , Tthey must also be open in the− relative (d) There exists a non-empty, proper, clopen subset 2 − 1 of X. topology of A. By a connected subset is meant a subset of X that Following are some useful properties of con- is connected when provided with its relative topol- nected spaces. ogy making it a subspace of X. Thus any connected subset of a topological space must necessarily be (c1) The closure of any connected subspace of a contained in any clopen set that might intersect it: space is connected. In general, every B satis- if C and H are respectively connected and clopen fying subsets of X such that C H = , then C H be- A B Cl(A) cause C H is a non-empty clop6 ∅en set in C⊂ which ⊆ ⊆ must contain C because CTis connected. is connected. Thus any subset of X formed For Ttesting whether a subset of a topological from A by adjoining to it some or all of its space is connected, the following relativized form of adherents is connected so that a topologi- (a)–(d) is often useful. cal space with a dense connected subset is connected. Lemma A.3.1. A subset A of X is disconnected iff (c2) The union of any class of connected subspaces there are disjoint open sets U and V of X satisfying of X with nonempty intersection is a con- U A = = V A such that A U V , nected subspace of X. 6 ∅ 6 ⊆ (c3) A topological space is connected iff there is a with U V A = T T ∅ S covering of the space consisting of connected T T (A.50) sets with nonempty intersection. Connected- or there are disjoint closed sets E and F of X ness is a topological property: Any space satisfying homeomorphic to a connected space is itself connected. E A = = F A such that A E F , 6 ∅ 6 ⊆ (c4) If H1 and H2 is a separation of X and A is any with E F A = . connected subset A of X, then either A H1 T T ∅ S ⊆ or A H2. T T (A.51) ⊆ Thus A is disconnected iff there are disjoint clopen While the real line R is connected, a subspace subsets in the relative topology of A that cover A. of R is connected iff it is an interval in R. December 3, 2003 12:13 00851

Toward a Theory of Chaos 3223

X − A

1. Donor 2. Selﬁsh (Closed) 3. Neutral XXX

1. Donor

AAopen A

X − A open X − A open X − A open

2. Selﬁsh A (Closed)

A Aopen A

X X X

3. Neutral

A A open A

Der(A)BBdyX−A(A) Der(X − A) dyA(X−A)

Fig. 22. Classification of a subset A of X relative to the topology of X. The derived set of A may intersect both A and X A (row 3), may be entirely in A (row 2), or may be wholly in X A (row 1). A is closed iff Bdy(A) A (row 2), open − − ⊆ iff Bdy(A) X A (column 2), and clopen iff Bdy(A) = when the derived sets of both A and X A are contained in the ⊆ − ∅ − respective sets. An open set, beside being closed, may also be neutral or donor.

The important concept of total disconnected- ply must not be contained in any other connected ness introduced below needs the following subset of X. Components can be constructively de- ﬁned as follows: Let x X be any point. Consider ∈ Deﬁnition A.3.2 (Component). A component C ∗ the collection of all connected subsets of X to which x belongs. Since x is one such a set, the collection of a space X is a maximally (with respect to { } inclusion) connected subset of X. is non-empty. As the intersection of the collection is non-empty, its union is a non-empty connected 1 set C. This is the largest connected set containing Thus a component is a connected subspace which x and is therefore a component containing x and we is not properly contained in any larger connected have subspace of X. The maximal element need not be unique as there can be more than one component of (C1) Let x X. The unique component of X con- ∈ a given space and a “maximal” criterion rather than taining x is the union of all the connected sub- “maximum” is used as the component that need sets of X that contain x. Conversely any non- not contain every connected subset of X; it sim- empty connected subset A of X is contained December 3, 2003 12:13 00851

3224 A. Sengupta

in that unique component of X to which each Table 7. Separation properties of some useful of the points of A belong. Hence a topological spaces. space is connected iff it is the unique compo- Space T T T nent of itself. 0 1 2 (C2) Each component C ∗ of X is a closed set of X: Discrete X X X ∗ Indiscrete By property (c1) above, Cl(C ) is also con- × × × nected and from C∗ Cl(C∗) it follows that R, standard X X X ∗ ∗ ⊆ left/right ray X C = Cl(C ). Components need not be open × × sets of X: an example of this is the space of ra- Infinite cofinite X X × Q Uncountable cocountable X X tionals in reals in which the components are × the individual points which cannot be open in x-inclusion/exclusion X × × R A-inclusion/exclusion ; see Example 2 below. × × × (C3) Components of X are equivalence classes of (X, ) with x y iff they are in the same comp∼onent: while∼ reflexivity and symmetry are obvious enough, transitivity follows be- (2, 3) can be enlarged into bigger connected subsets cause if x, y C and y, z C with C , ∈ 1 ∈ 2 1 of X. C2 connected subsets of X, then x and z As connected spaces, the empty set and the sin- are in the set C1 C2 which is connected by gleton are considered to be degenerate and any con- property c(2) above as they have the point y nected subspace with more than one point is non- S in common. Components are connected dis- degenerate. At the opposite extreme of the largest joint subsets of X whose union is X (i.e. they possible component of a space X which is X itself, form a partition of X with each point of X are the singletons x for every x X. This leads contained in exactly one component of X) to the extremely imp{ }ortant notion ∈of a such that any connected subset of X can be contained in only one of them. Because Definition A.3.3 (Totally disconnected space). A a connected subspace cannot contain in it space X is totally disconnected if every pair of dis- any clopen subset of X, it follows that every tinct points in it can be separated by a disconnec- clopen connected subspace must be a compo- tion of X. nent of X.

Even when a space is disconnected, it is always X is totally disconnected iff the components in possible to decompose it into pairwise disjoint con- X are single points with the only nonempty connected subsets. If X is a discrete space this is the nected subsets of X being the one-point sets: If only way in which X may be decomposed into con- x = y A X are distinct points of a sub- 6 ∈ ⊆ nected pieces. If X is not discrete, there may be set A of X then A = (A H1) (A H2), where other ways of doing this. For example, the space X = H1 H2 with x H1 and y H2 is a discon- ∈ T S∈ T nection of X (it is possible to choose H1 and H2 in X = x R : (0 x 1) (2 < x < 3) S { ∈ ≤ ≤ ∨ } this manner because X is assumed to be totally dis- has the following distinct decomposition into three connected), is a separation of A that demonstrates connected subsets: that any subspace of a totally disconnected space 1 1 7 7 with more than one point is disconnected. X = 0, , 1 2, , 3 2 2 3 3 A totally disconnected space has interesting physically appealing separation properties in terms S∞ S S 1 1 of the (separated) Hausdorff spaces; here a topo- X = 0 , (2, 3) { } n + 1 n logical space X is Hausdorff, or T2, iff each two n=1 ! S [ S distinct points of X can be separated by disjoint X = [0, 1] (2, 3) . neighborhoods, so that for every x = y X, there 6 ∈ Intuition tells usS that only in the third of these de- are neighborhoods M x and N y such that compositions have we really broken up X into its M N = . This means∈ Nthat for an∈yNtwo distinct connected pieces. What distinguishes the third from points x = ∅y X, it is impossible to find points that the other two is that neither of the pieces [0, 1] or areTarbitrarily6 ∈ close to both of them. Among the December 3, 2003 12:13 00851

Toward a Theory of Chaos 3225

properties of Hausdorff spaces, the following need It should be noted that that as none of the prop- to be mentioned. erties (H1)–(H3) need neighborhoods of both points simultaneously, it is sufficient for X to be T for the (H1) X is Hausdorff iff for each x X and any 1 conclusions to remain valid. point y = x, there is a neighborho∈ od N of x From its definition it follows that any totally such that6 y Cl(N). This leads to the sig- disconnected space is a Hausdorff space and is there- nificant result6∈ that for any x X the closed fore both T and T spaces as well. However, if a singleton ∈ 1 0 Hausdorff space has a base of clopen sets then it is x = Cl(N) totally disconnected; this is so because if x and y { } are distinct points of X, then the assumed property N\∈Nx of x H M for every M and some clopen is the intersection of the closures of any local ∈ ⊆ ∈ Nx base at that point, which in the language of set M yields X = H (X H) as a disconnection of X that separates x and y− X H; note that the nets and filters (Appendix A.1) means that a S ∈ − net in a Hausdorff space cannot converge to assumed Hausdorffness of X allows M to be chosen more than one point in the space and the ad- so as not to contain y. herent set adh( ) of the neighborhood filter x Example A.3.1 at x is the singletonN x . { } (H2) Since each singleton is a closed set, each fi- (1) Every indiscrete space is connected; every sub- nite set in a Hausdorff space is also closed in set of an indiscrete space is connected. Hence X. Unlike a cofinite space, however, there can if X is empty or a singleton, it is connected. A clearly be infinite closed sets in a Hausdorff discrete space is connected iff it is either empty space. or is a singleton; the only connected subsets in (H3) Any point x in a Hausdorff space X is a clus- a discrete space are the degenerate ones. This is ter point of A X iff every neighborhood an extreme case of lack of connectedness, and a of x contains infinitely⊆ many points of A, a discrete space is the simplest example of a total fact that has led to our mental conditioning disconnected space. of the points of a (Cauchy) sequence piling up (2) Q, the set of rationals considered as a subspace in neighborhoods of the limit. Thus suppose of the real line, is (totally) disconnected because for the sake of argument that although some all rationals larger than a given irrational r is a neighborhood of x contains only a finite num- clopen set in Q, and ber of points, x is nonetheless a cluster point Q = ( , r) Q Q (r, ) of A. Then there is an open neighborhood U −∞ ∞ of x such that U (A x ) = x1, . . . , xn is r is an irrational\ [ \ a finite closed set of X−{not} con{taining x, and} is the union of two disjoint clopen sets in the U (X x , . . .T, x ) being the intersection 1 n relative topology of Q. The sets ( , r) Q of two op−en{ sets, is an}open neighborhood of x and Q (r, ) are clopen in Q because−∞ neither∩ notTintersecting A x implying thereby that contains∩ a cluster∞ point of the other. Thus for x Der(A); infact−U{ } (X x , . . . , x ) is 1 n example, any neighborhood of the second must simply6∈ x if x A or belongs− {to Bdy }(A) X−A contain the irrational r in order to be able to cut when x{ }X ∈A. ConTversely if every neigh- the first which means that any neighborhood borhood∈of a−point of X intersects A in in- of a point in either of the relatively open sets finitely many points, that point must belong cannot be wholly contained in the other. The to Der(A) by definition. only connected sets of Q are one point subsets consisting of the individual rationals. In fact, a Weaker separation axioms than Hausdorffness connected piece of Q, being a connected subset are those of T0, respectively T1, spaces in which of R, is an interval in R, and a nonempty in- for every pair of distinct points at least one, re- terval cannot be contained in Q unless it is a spectively each one, has some neighborhood not singleton. It needs to be noted that the individ- containing the other; the following table is a list- ual points of the rational line are not (cl)open ing of the separation properties of some useful because any open subset of R that contains a spaces. rational must also contain others different from December 3, 2003 12:13 00851

3226 A. Sengupta

it. This example shows that a space need not spaces, the proof of which uses this contrapositive be discrete for each of its points to be a com- characterization of compactness. ponent and thereby for the space to be totally Theorem A.2.1. A topological space X is compact disconnected. iff each class of closed subsets of X with finite in- In a similar fashion, the set of irrationals tersection property has non-empty intersection. is (totally) disconnected because all the irrationals larger than a given rational is an exam- Proof. Necessity. Let X be a compact space. Let ple of a clopen set in R Q. = Fα α∈D be a collection of closed subsets of X (3) The p-inclusion (A-inclusion)− topology is con- F { } with finite FIP, and let = X Fα α∈D be the nected; a subset in this topology is connected G { − N} corresponding open sets of X. If Gi i=1 is a non- iff it is degenerate or contains p. For, a sub- { } N empty finite subcollection from , then X Gi i=1 set inherits the discrete topology if it does is the corresponding non-emptyGfinite sub{ collection− } not contain p, and p-inclusion topology if it of . Hence from the assumed finite intersection contains p. propFerty of , it must be true that (4) The cofinite (cocountable) topology on an infi- F N N nite (uncountable) space is connected; a subset X G = (X G ) (DeMorgan’s Law) in a cofinite (cocountable) space is connected iff − i − i i=1 i=1 it is degenerate or infinite (countable). [ \ = , (5) Removal of a single point may render a con- 6 ∅ nected space disconnected and even totally so that no finite subcollection of can cover X. Compactness of X now implies thatG too cannot disconnected. In the former case, the point G removed is called a cut point and in the sec- cover X and therefore ond, it is a dispersion point. Any real number F = (X G ) = X G = . α − α − α 6 ∅ is a cut point of R and it does not have any α α α \ \ [ dispersion point only. The proof of the converse is a simple exercise of re- (6) Let X be a topological space. Considering com- versing the arguments involving the two equations ponents of X as equivalence classes by the in the proof above. equivalence relation with Q : X X/ denoting the quotient∼map, X/ is totally→ dis-∼ Our interest in this theorem and its proof lies in connected: As Q−([x]) is connected∼ for each the following corollary — which essentially means [x] X/ in a component class of X, and that for every filter on a compact space the adher- F as an∈y open∼ or closed subset A X/ is con- ent set adh( ) is not empty — from which it follows F nected iff Q−(A) is open or closed,⊆ it ∼must fol- that every net in a compact space must have a con- low that A can only be a singleton. vergent subnet.

The next notion of compactness in topological Corollary. A space X is compact iff for every class spaces provides an insight of the role of non-empty = (A ) of nonempty subsets of X with FIP, A α adherent sets of filters that lead in a natural fash- adh( ) = Cl(A ) = . A Aα∈A α 6 ∅ ion to the concept of attractors in the dynamical systems theory that we take up next. The proTof of this result for nets given by the next theorem illustrates the general approach in Definition A.3.4 (Compactness). A topological such cases which is all that is basically needed in space X is compact iff every open cover of X con- dealing with attractors of dynamical systems; com- tains a finite subcover of X. pare Theorem A.1.3. Theorem A.3.2. A topological space X is compact This definition of compactness has an useful iff each net in X adheres to X. equivalent contrapositive reformulation: For any given collection of open sets of X if none of its Proof. Necessity. Let X be a compact space, χ : finite subcollections cover X, then the entire col- D X a net in X, and R the residual of α in the → α lection also cannot cover X. The following theorem directed set D. For the filter-base (F χ(Rα))α∈D of is a statement of the fundamental property of com- nonempty, decreasing, nested subsetsBof X associ- pact spaces in terms of adherences of filters in such ated with the net χ, compactness of X requires from December 3, 2003 12:13 00851

Toward a Theory of Chaos 3227

αδ Cl(χ(Rα) χ(Rδ) = , that the uncountably compactness of subspaces: A subspace K of a topo- intersecting subset⊇ 6 ∅ logical space X is compact iff each open cover of K T in X contains a finite cover of K. adh(F χ) := Cl(χ(Rα)) A proper understanding of the distinction be- B D α\∈ tween compactness and closedness of subspaces — which often causes much confusion to the non- of X be non-empty. If x adh(F χ) then because ∈ B specialist — is expressed in the next two theorems. x is in the closure of χ(Rβ), it follows from Eq. (20) 33 As a motivation for the first that establishes that that N χ(Rβ) = for every N x, β D. Hence χ(γ) N for6 ∅some γ β so that∈ Nx adh(∈χ); not every subset of a compact space need be com- see Eq. T(A.16).∈ ∈ pact, mention may be made of the subset (a, b) of R Sufficiency. Let χ be a net in X that adheres the compact closed interval [a, b] in . at x X. From any class of closed subsets of Theorem A.3.3. A closed subset F of a compact ∈ F X with FIP, construct as in the proof of Theo- space X is compact. rem A.1.4, a decreasing nested sequence of closed subsets C = F : F and consider Proof. Let be an open cover of F so that an β αβ∈D{ α α ∈ F} G the directed set DC = (C , β) : (β D)(x open cover of X is (X F ), which because T β { β ∈ β ∈ G − Cβ) with its natural direction (A.11) to define the of compactness of X contains a finite subcover . } Then (X F ) is aSfinite collection of thatU net χ(Cβ, β) = xβ in X; see Definition A.1.10. From U − − G the assumed adherence of χ at some x X, it covers F . follows that N F = for every N ∈ and x It is not true in general that a compact subset F . Hence x belongs6 ∅to the closed set∈FNso that of a space is necessarily closed. For example, in an x ∈ adh(F ); seeTEq. (A.24). Hence X is compact. infinite set X with the cofinite topology, let F be ∈ F an infinite subset of X with X F also infinite. Then although F is not closed in−X, it is neverthe- Using Theorem A.1.5 that specifies a definite less compact because X is compact. Indeed, let criterion for the adherence of a net, this theorem be an open cover of X and choose any non-emptGy reduces to the useful formulation that a space is G . If G = X then G is the required fi- compact iff each net in it has some convergent sub- 0 0 0 nite∈coGver of X. If this is not{ the} case, then because net. An important application is the following: Since n X G0 = xi i=1 is a finite set, there is a Gi every decreasing sequence (Fm) of nonempty sets − { } ∈ G M with xi Gi for each 1 i n, and therefore has FIP (because Fm = FM for every finite n ∈ ≤ ≤ m=1 Gi is the finite cover that demonstrates the M), every decreasing sequence of nonempty closed { }i=0 T compactness of the cofinite space X. Compactness subsets of a compact space has nonempty intersec- of F now follows because the subspace topology on tion. For a complete metric space this is known as F is the induced cofinite topology from X. The dis- the Nested Set Theorem, and for [0, 1] and other tinguishing feature of this topology is that it, like compact subspaces of R as the Cantor Intersection 34 the cocountable, is not Hausdorff: If U and V are Theorem. any two nonempty open sets of X, then they can- For subspaces A of X, it is the relative topology not be disjoint as the complements of the open sets that determines as usual compactness of A; however can only be finite and if U V were to be indeed the following criterion renders this test in terms of empty, then the relative topology unnecessary and shows that T the topology of X itself is sufficient to determine X = X = X (U V ) = (X U) (X V ) − ∅ − − − \ [ 33 This is of course a triviality if we identify each χ(Rβ) (or F in the proof of the converse that follows) with a neighborhood N of X that generates a topology on X. 34 Nested-set theorem. If (En) is a decreasing sequence of nonempty, closed, subsets of a complete metric space (X, d) such that limn→∞ dia(En) = 0, then there is a unique point ∞ x En . ∈ n \=0 The uniqueness arises because the limiting condition on the diameters of En imply, from property (H1), that (X, d) is a Hausdorff space. December 3, 2003 12:13 00851

3228 A. Sengupta

would be a finite set. An immediate fallout of this is Vy is a neighborhood of x and the intersection is that in an infinite cofinite space, a sequence (xi)i∈N over finitely many points y of A. To prove that K is (and even a net) with xi = xj for i = j behaves in closed in X it is enough to show that V is disjoint an extremely unusual way:6 It conver6 ges, as in the from K: If there is indeed some z V K then z ∈ indiscrete space, to every point of the space. Indeed must be in some Uy for y A. But as z V it is ∈ T ∈ if x X, where X is an infinite set provided with also in Vy which is impossible as Uy and Vy are to its cofinite∈ topology, and U is any neighborhood of be disjoint. This last part of the argument in fact x, any infinite sequence (xi)i∈N in X must be even- shows that if K is a compact subspace of a Haus- tually in U because X U is finite, and ignoring dorff space X and x / K, then there are disjoint of the initial set of its v−alues lying in X U in no open sets U and V of∈X containing x and K. way alters the ultimate behavior of the−sequence (note that this implies that the filter induced on The last two theorems may be combined to give X by the sequence agrees with its topology). Thus the obviously important xi x for any x X is a reflection of the fact that there→ are no small∈neighborhoods of any point of X Corollary. In a compact Hausdorff space, closed- with every neighborhood being almost the whole of ness and compactness of its subsets are equivalent X, except for a null set consisting of only a finite concepts. number of points. This is in sharp contrast with Hausdorff spaces where, although every finite set is In the absence of Hausdorffness, it is not pos- also closed, every point has arbitrarily small neigh- sible to conclude from the assumed compactness of borhoods that lead to unique limits of sequences. A the space that every point to which the net may corresponding result for cocountable spaces can be converge actually belongs to the subspace. found in Example A.1.2, continued. Definition A.3.5. A subset D of a topological This example of the cofinite topology motivates space (X, ) is dense in X if Cl(D) = X. Thus the following “converse” of the previous theorem. the closureUof D is the largest open subset of X, Theorem A.3.4. Every compact subspace of a and every neighborhood of any point of X contains Hausdorff space is closed. a point of D not necessarily distinct from it; refer to the distinction between Eqs. (20) and (22). Proof. Let K be a non-empty compact subset of X, Fig. 23, and let x X K. Because of the sep- Loosely, D is dense in X iff every point of X aration of X, for every∈ y− K there are disjoint has points of D arbitrarily close to it. A self-dense ∈ open subsets Uy and Vy of X with y Uy, and (dense in itself ) set is a set without any isolated x V . Hence U is an open cover ∈for K, and points; hence A is self-dense iff A Der(A). A y y y∈K ⊆ from∈ its compactness{ } there is a finite subset A of closed self-dense set is called a perfect set so that a closed set A is perfect iff it has no isolated points. K such that K y∈A Uy with V = y∈A Vy an open neighborho⊆od of x; V is open because each Accordingly S T A is perfect A = Der(A) , ⇔ means that the closure of a set without any isolated points is a perfect set.

Theorem A.3.5. The following are equivalent .

statements.

¢ (1) D is dense in X. (2) If F is any closed set of X with D F, then

F = X; thus the only closed superset⊆ of D is

. X.

(3) Every nonempty (basic) open set of X cuts D; ¡£¢¥¤§¦©¨ thus the only open set disjoint from D is the empty set . ∅ Fig. 23. Closedness of compact subsets of a Hausdorﬀ space. (4) The exterior of D is empty. December 3, 2003 12:13 00851

Toward a Theory of Chaos 3229

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(a)(a)(a)(a)AAAisAisisolatedisisolatedisolatedisolated (b(b(b))(b)A)AAisisisnisnnwwdnwdwdd ((c)c)(c)(c)CCisisCisCanCisCaCnatorntaorntortor C Fig. 24. Shows the distinction between isolated, nowhere dense and Cantor sets. Topologically, the Cantor set can be described as a perfect, nowhere dense, totally disconnected and compact subset of a space. (b) The closed nowhere dense set Cl(A) is the boundary of its open complement. Here downward and upward inclined hatching denote respectively Bdy (X A) and A − BdyX−A(A).

Proof. (3) If U indeed is a non-empty open set of closure, that is A Cl(X Cl(A)). In particu- X with U D = , then D X U = X leads lar a closed subset⊆A is no−where dense in X iff to the contradiction∅ X = Cl(⊆D) −Cl(6X U) = A = Bdy(A), that is iff it contains no open set. ⊆ − X U = TX, which also incidentally proves (2). (2) From M N Cl(M) Cl(N) it follows, − 6 From (3) it follows that for any open set U of with M =⊆X ⇒Cl(A) and⊆N = X A, that X, Cl(U) = Cl(U D) because if V is any open a nowhere dense− set is residual, but −a residual neighborhood of x Cl(U) then V U is a non- set need not be nowhere dense unless it is also ∈ empty open set ofTX that must cut D so that closed in X. V (U D) = implies x Cl(U T D). Finally, (3) Since Cl(Cl(A)) = Cl(A), Cl(A) is nowhere 6 ∅ ∈ Cl(U D) Cl(U) completes the proof. dense in X iff A is. T T ⊆ T (4) For any A X, both BdyA(X A) := Cl(X DefinitionT A.3.6. (a) A set A X is said to be ⊆ − − ⊆ A) A and BdyX−A(A) := Cl(A) (X nowhere dense in X if Int(Cl(A)) = and residual A) are residual sets and as Fig. 22 shows− in X if Int(A) = . ∅ T T ∅ BdyX (A) = BdyX−A(A) BdyA(X A) is the A is nowhere dense in X iff union of these two residual sets. When− A is Bdy(X Cl(A)) = Bdy(Cl(A)) = Cl(A) closed (or open) with XS its boundary, con- − sisting of the only component Bdy (X A) so that A − (or BdyX−A(A)) as shown by the second row Cl(X Cl(A)) = (X Cl(A)) Cl(A) = X − − (or column) of the figure, being a closed set from which it follows that [ of X is also nowhere dense in X; in fact a closed nowhere dense set is always the bound- A is nwd in X X Cl(A) is dense in X ⇔ − ary of some open set. Otherwise, the bound- and ary components of the two residual parts — A is residual in X X A is dense in X . as in the donor–donor, donor–neutral, neutral– ⇔ − donor and neutral–neutral cases — need not Thus A is nowhere dense iff Ext(A) := X be individually closed in X (although their Cl(A) is dense in X, and in particular, a closed set− union is) and their union is a residual set that is nowhere dense in X iff its complement is op11en1 need not be nowhere dense in X: the union dense in X with open-denseness being complimen- of two nowhere dense sets is nowhere dense tarily dual to closed-nowhere denseness. The ratio- but the union of a residual and a nowhere nals in reals is an example of a set that is residual dense set is a residual set. One way in which but not nowhere dense. The following are readily a two-component boundary can be nowhere verifiable properties of subsets of X. dense is by having Bdy (X A) Der(A) or A − ⊇ (1) A set A X is nowhere dense in X iff it is BdyX−A(A) Der(X A), so that it is effec- contained⊆in its own boundary, iff it is con- tively in one piece⊇ rather−than in two, as show in tained in the closure of the complement of its Fig. 24(b). December 3, 2003 12:13 00851

3230 A. Sengupta

Theorem A.3.6. A is nowhere dense in X iff each x(0) x(0) 1 2 non-empty open set of X has a non-empty open sub- ..C0 x(1) x(1) set disjoint from Cl(A). 2 3 .. C1 x(2) x(2) x(2) x(2) Proof. 2 3 6 7 If U is a non-empty open set of X, then .. .. C2 (3) (3) (3) (3) U0 = U Ext(A) = as Ext(A) is dense in X; U0 x x x x ∩ 6 ∅ 3 7 10 14 C is the open subset that is disjoint from Cl(A). It .. ..3 clearly follows from this that each non-empty open C set of X has a non-empty open subset disjoint from 4 a nowhere dense set A. C What this result (which follows just from the Fig. 25. Construction of the classical 1/3-Cantor set. The definition of nowhere dense sets) actually means is 1 end points of C3, for example, in increasing order are: 0, 27 ; that no point in Bdy (A) can be isolated in it. 2 1 2 7 8 1 2 19 20 7 8 25 26 | | X−A , ; , ; , ; , ; , ; , ; , 1 . Ci is | 27 9 | | 9 27 | | 27 3 | | 3 27 | | 27 9 | | 9 27 | | 27 | the union of 2i pairwise disjoint closed intervals each of length −i ∞ Corollary. A is nowhere dense in X iff Cl(A) does 3 and the non-empty infinite intersection = Ci C ∩i=0 not contain any non-empty open set of X iff any is the adherent Cantor set of the filter-base of closed sets C , C , C , . . . . nonempty open set that contains A also contains its { 0 1 2 } closure.

Example A.3.2. Each finite subset of Rn is number — it follows that both rationals and irra- nowhere dense in Rn; the set 1/n ∞ is nowhere tionals belong to the Cantor set. { }n=1 dense in R. The Cantor set is nowhere dense in ( 1) is totally disconnected. If possible, let C [0, 1] because every neighborhood of any point in C Chave a component containing points a and Cb must contain, by its very construction, a point with a < b. Then [a, b] [a, b] C for C i with 1 in its ternary representation. That the in- all i. But this is impossible⊆ C ⇒because⊆we may terior and the interior of the closure of a set are choose i large enough to have 3−i < b a not necessarily the same is seen in the example so that a and b must belong to two differ-− of the rationals in reals: The set of rational num- ent members of the pairwise disjoint closed bers Q has empty interior because any neighbor- 2i subintervals each of length 3−i that consti- hood of a rational number contains irrational num- tutes Ci. Hence bers (so also is the case for irrational numbers) and [a, b] is not a subset of any C [a, b] R = Int(Cl(Q)) Int(Q) = justifies the notion of i ⇒ a nowhere dense⊇set. ∅ is not a subset of . C The following properties of can be taken to ( 2) is perfect so that for any x every C C ∈ C define any subset of a topologicalC space as a Can- neighborhood of x must contain some other tor set; set-theoretically it should be clear from its point of . Supposing to the contrary that the C classical middle-third construction that the Cantor singleton x is an open set of , there must { } C set consists of all points of the closed interval [0, be an ε > 0 such 1that in the usual topology 1] whose infinite triadic (base 3) representation, ex- of R pressed so as not to terminate with an infinite string x = (x ε, x + ε) . (A.52) of 1’s, does not contain the digit 1. Accordingly, any { } C − end point of the infinite set of closed intervals whose Choose a positive\integer i large enough to −i intersection yields the Cantor set, is represented by satisfy 3 < ε. Since x is in every Ci, it must a repeating string of either 0 or 2 while a non end be in one of the 2i pairwise disjoint closed in- point has every other arbitrary collection of these tervals [a, b] (x ε, x + ε) each of length −i ⊂ − two digits. Recalling that any number in [0, 1] is a 3 whose union is Ci. As [a, b] is an interval, rational iff its representation in any base is termi- at least one of the end points of [a, b] is dif- nating or recurring — thus any decimal that neither ferent from x, and since an end point belongs repeats or terminates but consists of all possible se- to , (x ε, x + ε) must also contain this quences of all possible digits represents an irrational poinC tCthereb∩ −y violating Eq. (A.52). December 3, 2003 12:13 00851

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( 3) is nowhere dense because each neighbor- where λ(ν) is the usual combination coefficient C Chood of any point of intersects Ext( ); see of the solutions of the homogeneous and non- Theorem A.3.6. C C homogeneous parts of a linear equation, ( ) is a P · ( 4) is compact because it is a closed subset principal value and δ(x) the Dirac delta, to lead C Ccontained in the compact subspace [0, 1] of to the full-range 1 µ 1 solution valid for − ≤ ≤ R, see Theorem A.3.3. The compactness of < x < −∞ ∞ [0, 1] follows from the Heine-Borel Theorem Φ(x, µ) = a(ν )e−x/ν0 φ(µ, ν ) which states that any subset of the real line 0 0 is compact iff it is both closed and bounded + a( ν )ex/ν0 φ( ν , µ) − 0 − 0 with respect to the Euclidean metric on R. 1 + a(ν)e−x/ν φ(µ, ν)dν (A.56) Compare ( 1) and ( 2) with the essentially Z−1 C C similar arguments of Example A.3.1(2) for the sub- of the one-speed neutron transport equation (A.53). space of rationals in R. Here the real ν0 and ν satisfy respectively the integral constraints cν ν + 1 A.4. Neutron Transport Theory 0 ln 0 = 1, ν > 1 2 ν 1 | 0| This section introduces the reader to the basics of 0 − the linear neutron transport theory where graphi- cν 1 + ν λ(ν) = 1 ln , ν [ 1, 1] , cal convergence approximations to the singular dis- − 2 1 ν ∈ − tributions, interpreted here as multifunctions, led − with to the study of this paper. The one-speed (that is cν 1 mono-energetic) neutron transport equation in one φ(µ, ν ) = 0 0 2 ν µ dimension and plane geometry, is 0 − following from Eq. (A.55). ∂Φ(x, µ) c 1 µ + Φ(x, µ) = Φ(x, µ0)dµ0 , It can be shown [Case & Zweifel, 1967] that the ∂x 2 Z−1 eigenfunctions φ(ν, µ) satisfy the full-range orthog- 0 < c < 1, 1 µ 1 onality condition − ≤ ≤ (A.53) 1 µφ(ν, µ)φ(ν0, µ)dµ = N(ν)δ(ν ν0) , − where x is a non-dimensional physical space variable Z−1 that denotes the location of the neutron moving in where the odd normalization constants N are given a direction θ = cos−1(µ), Φ(x, µ) is a neutron den- by sity distribution function such that Φ(x, µ)dxdµ is 1 the expected number of neutrons in a distance dx N( ν ) = µφ2( ν , µ)dµ for ν > 1 0 0 | 0| about the point x moving at constant speed with Z−1 their direction cosines of motion in dµ about µ, cν3 c 1 = 0 , and c is a physical constant that will be taken to 2 ν2 1 − ν2 satisfy the restriction shown above. Case’s method 0 − 0 starts by assuming the solution to be of the form and −x/µ 2 Φν(x, µ) = e φ(µ, ν) with a normalization inte- 2 πcν 1 N(ν) = ν λ (ν) + for ν [ 1, 1] . gral constraint of −1 φ(µ, ν)dµ = 1 to lead to the 2 ∈ − simple equation R With a source of particles ψ(x0, µ) located at x = cν (ν µ)φ(µ, ν) = (A.54) x0 in an infinite medium, Eq. (A.56) reduces to the − 2 boundary condition, with µ, ν [ 1, 1], ∈ − for the unknown function φ(ν, µ). Case then sug- −x0/ν0 ψ(x0, µ) = a(ν0)e φ(µ, ν0) gested, see [Case & Zweifel, 1967], the non-simple x0/ν0 complete solution of this equation to be + a( ν0)e φ( ν0, µ) − − cν 1 1 φ(µ, ν) = + λ(v)δ(v µ) , (A.55) + a(ν)e−x0/νφ(µ, ν)dν (A.57) 2 P ν µ − −1 − Z December 3, 2003 12:13 00851

3232 A. Sengupta

for the determination of the expansion coefficients 1 W (µ)φ(µ, ν0)φ(µ, ν)dµ a( ν0), a(ν) . Use of the above orthogonal- ν∈[−1,1] 0 − ityintegrals{ then} lead to the complete solution of Z cν0 the problem to be = (ν + ν)φ(ν0, ν)X( ν) 2 0 − − ex0/ν 1 a(ν) = µψ(x0, µ)φ(µ, ν)dµ , where the half-range weight function W (µ) is N(ν) Z−1 defined as ν = ν0 or ν [ 1, 1] . cµ ∈ − W (µ) = (A.61) 2(1 c)(ν + µ)X( µ) For example, in the infinite-medium Greens func- − 0 − tion problem with x = 0 and ψ(x , µ) = δ(µ 0 0 − in terms of the X-function µ0)/µ, the coefficients are a( ν0) = φ(µ0, ν0)/ N( ν ) when ν = ν , and a(ν) = φ(µ , ν)/N(ν) 0 0 0 c 1 ν cν2 X( µ) = exp 1 + ln(ν + µ)dν , for ν [ 1, 1]. − − 2 N(ν) 1 ν2 F∈or −a half-space 0 x < , the obvious reduc- ( Z0 − ) tion of Eq. (A.56) to ≤ ∞ 0 µ 1 , ≤ ≤

−x/ν0 Φ(x, µ) = a(ν0)e φ(µ, ν0) that is conveniently obtained from a numerical so- 1 lution of the nonlinear integral equation + a(ν)e−x/ν φ(µ, ν)dν (A.58) cµ 1 ν2(1 c) ν2 Z0 Ω( µ)=1 0 − − dν − − 2(1 c) (ν2 ν2)(µ+ν)Ω( ν) with boundary condition, µ, ν [0, 1], − Z0 0 − − ∈ (A.62) −x0/ν0 ψ(x0, µ) = a(ν0)e φ(µ, ν0) to yield 1 −x0/ν + a(ν)e φ(µ, ν)dν , (A.59) Ω( µ) Z0 X( µ) = − , − µ + ν √1 c leads to an infinitely more difficult determination of 0 − the expansion coefficients due to the more involved and X( ν0) satisfy nature of the orthogonality relations of the eigenfunctions in the half-interval [0, 1] that now reads ν2(1 c) 1 X(ν )X( ν ) = 0 − − . for ν, ν0 [0, 1] [Case & Zweifel, 1967] 0 − 0 2(1 c)v2(ν2 1) ∈ − 0 0 − 1 0 Two other useful relations involving the W -function W (µ)φ(µ, ν )φ(µ, ν)dµ 1 0 are given by 0 W (µ)φ(µ, ν0)dµ = cν0/2 and Z 1 W (ν)N(ν) W (µ)φ(µ, ν)dµ = cν/2. = δ(ν ν0) 0 R ν − The utility of these full- and half-range orthog- 1 Ronality relations lie in the fact that a suitable class W (µ)φ(µ, ν0)φ(µ, ν)dµ = 0 of functions of the type that is involved here can al- Z0 ways be expanded in its terms, see [Case & Zweifel, 1 1967]. An example of this for a full-range problem W (µ)φ(µ, ν0)φ(µ, ν)dµ − has been given above; we end this introduction to Z0 = cνν X( ν )φ(ν, ν ) (A.60) the generalized — traditionally known as singular in 0 − 0 − 0 1 neutron transport theory — eigenfunction method W (µ)φ(µ, ν )φ(µ, ν )dµ with two examples of half-range orthogonality inte- 0 0 Z0 grals to the half-space problems A and B of Sec. 5. cν0 2 = X( ν0) ∓ 2 Problem A (The Milne Problem). In this case 1 there is no incident flux of particles from outside W (µ)φ(µ, ν0)φ(µ, ν)dµ 0 − the medium at x = 0, but for large x > 0 the Z 2 c νν0 neutron distribution inside the medium behaves = X( ν) x/ν0 4 − like e φ( ν0, µ). Hence the boundary condition − December 3, 2003 12:13 00851

Toward a Theory of Chaos 3233

(A.59) at x = 0 reduces to which leads, using the integral relations satisfied by W , to the expansion coefficients φ(µ, ν ) = a (ν )φ(µ, ν ) − − 0 A 0 0 a (ν ) = 2/cν X(v ) B 0 − 0 0 1 1 (A.64) + aA(ν)φ(µ, ν)dν µ 0 a (ν) = (1 c)ν(ν + ν)X( ν) . ≥ B 0 Z0 N(ν) − − Use of the fourth and third equations of Eq. (A.60) where X( ν0) are related to Problem A as and the explicit relation Eq. (A.61) for W (µ) gives 2 respectively the coefficients 1 ν0 (1 c) 1 X(ν0) = − − 2 ν0 s2aA(ν0)(1 c)(ν 1) − 0 − X( ν0) aA(ν0) = − 2 X(v0) 1 aA(ν0) ν0 (1 c) 1 X( ν0) = −2 − . − ν0 2(1 c)(ν 1) 1 2 s 0 aA(ν) = c(1 c)ν νX( ν0)X( ν) . − − −N(ν) − 0 − − This brief introduction to the singular eigen- (A.63) function method should convince the reader of the great difficulties associated with half-space, half- The extrapolated end point z0 of Eq. (67) is re- range methods in particle transport theory; note lated to aA(ν0) of the Milne problem by aA(ν0) = that the X-functions in the coefficients above must exp( 2z0/ν0). be obtained from numerically computed tables. In − − contrast, full-range methods are more direct due to Problem B (The Constant Source Problem). Here the simplicity of the weight function µ, which sug- the boundary condition at x = 0 is gests the full-range formulation of half-range prob-

1 lems presented in Sec. 5. Finally it should be men- 1 = a (ν )φ(µ, ν ) + a (ν)φ(µ, ν)dν µ 0 tioned that this singular eigenfunction method is B 0 0 B ≥ Z0 based on the theory of singular integral equations.