A Concept of Homeomorphic Defect for Defining Mostly Conjugate

CHAOS 18, 013118 ͑2008͒

A concept of homeomorphic defect for deﬁning mostly conjugate dynamical systems ͒ ͒ Joseph D. Skufcaa and Erik M. Bolltb Department of Mathematics, Clarkson University, Potsdam, New York 13699-5815, USA ͑Received 11 September 2007; accepted 27 December 2007; published online 11 March 2008͒

A centerpiece of dynamical systems is comparison by an equivalence relationship called topological conjugacy. We present details of how a method to produce conjugacy functions based on a functional ﬁxed point iteration scheme can be generalized to compare dynamical systems that are not conjugate. When applied to nonconjugate dynamical systems, we show that the ﬁxed-point iteration scheme still has a limit point, which is a function we now call a “commuter”—a nonhomeomorphic change of coordinates translating between dissimilar systems. This translation is natural to the concepts of dynamical systems in that it matches the systems within the language of their orbit structures, meaning that orbits must be matched to orbits by some commuter function. We introduce methods to compare nonequivalent systems by quantifying how much the commuter function fails to be a homeomorphism, an approach that gives more respect to the dynamics than the traditional comparisons based on normed linear spaces, such as L2. Our discussion addresses a fundamental issue—how does one make principled statements of the degree to which a “toy model” might be representative of a more complicated system? © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2837397͔

A primary interest of this work will be to develop principled statements of the degree to which a “toy model” ciples and methods to compare dynamical systems when might be representative of a more complicated system. they are not necessarily equivalent (in the sense of conjugacy), but in a manner that respects the behavioral im- plications of conjugacy. That is, we extend a centerpiece I. INTRODUCTION notion in dynamical systems whose equivalence relation- The complex oscillations we see in the physical world ship called topological conjugacy can either judge two around us have been a subject of intense study in practically systems as the same, or not the same, a boolean answer. every branch of science and engineering, as documented in To this end, we detail methods to produce conjugacy many textbooks,1 review articles,2 and popular literature.3 functions based on a functional fixed-point iteration Since the beginnings of the field of dynamical systems by scheme. We then show how this same fixed-point iteration Henri Poincaré,4 characterizing a dynamical system has fo- can be generalized to compare dynamical systems that cused on examining the topological and geometric features are not conjugate. When applied to nonconjugate dy- of orbits, rather than focusing on the empirical details of the namical systems, we show that the fixed-point iteration solution of the dynamical system with respect to a specific scheme still has a limit point, which is a function we now coordinate system. One seeks to understand coordinate- call a “commuter”—a nonhomeomorphic change of coor- independent properties, such as the periodic orbit structure, dinates translating between dissimilar systems. This the count, and stability of periodic orbits. The question of translation is natural to the concepts of dynamical sys- whether two systems are dynamically the same has evolved tems in that it matches the systems within the language of into the modern notion of deciding if there is a conjugacy their orbit structures, meaning, in some sense, that orbits between them.5–9 must be matched to orbits by some commuter function. Since modeling is fundamental in science, we may ask, We introduce methods to compare nonequivalent systems “What do we mean by a model?” We would answer that a by quantifying how much the commuter function fails to model of the “true” system ͑physical perhaps͒ is a simpler be a homeomorphism, an approach that gives more re- system ͓perhaps of ordinary differential equations ͑ODEs͒, spect to the dynamics than the traditional comparisons for example͔ that is “descriptive” of some aspects of the 2 based on normed linear spaces, such as L . We coin the original system. A model should be a system that is somehow phrase “defect,” between two systems embedded in mea- easier to analyze. Although the model is only representative sure spaces, to describe relative measured failures of the of the true system, it might have been constructed with first commuter to be a homeomorphism. Our discussion ad- principles in mind and may teach us about the mechanisms dresses a fundamental issue—how does one make prin- of the true system. Given these notions, we tend to speak of a “toy model”—a dynamical system that is much “like” the ͒ a Electronic mail: [email protected]. “real” system. These subjective evaluations are assertions ͒ b Electronic mail: [email protected]. that the model is satisfactory, but fail to distinguish excellent

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FIG. 1. ͑Color͒ Conjugacy and commuter, shown as quadwebs. ͑Left͒ Quadwebs allow visualization of the commutative relationship of Eq. ͑1͒. The lower ͓ ͔ right panel shows a symmetric full shift tent map, g1 acting on space X= 0,1 . The upper right panel shows a graph of the conjugacy function, f, which maps ͓ ͔ ͓ ͔ ͓ ͔ X= 0,1 on the horizontal to Y = 0,1 on the vertical. The upper left panel shows a skew tent map g2 acting on Y = 0,1 , where we have oriented the graph by rotating the figure counterclockwise so that y= f͑x͒ lies in the domain of this graph. Similarly, the lower left is another copy of f, oriented to allow points ؠ ؠ ͒ ͑ x in the range of g1 to map to points y in the range of g2. The rectangles illustrate that the maps actually satisfy Eq. 1 , f g1 =g2 f.Inthisexample,the conjugacy is strictly increasing, yet its derivative is 0 almost everywhere, a Lebesgue singular function. In Appendix B, we discuss this singular property. ͑ ͒ ⌺Ј Right The two maps shown are obviously not conjugate, as there are a different number of humps; g1 requires a subshift of 2-symbols 2, and g2 requires ⌺Ј ͑ ͒ a subshift 4 of 4-symbols. So any f that satisfies Eq. 1 cannot be a homeomorphism. The “solution” f, found by fixed point iteration, we call a “commuter,” ؠ ؠ ͓ ͔→͓ ͔ −1 since it solves f g1 =g2 f. We see that as a function f : 0,1 0,1 , the commuter fails each of the properties of continuity, one-one-ness, onto-ness, and f fails to exist at certain points ͑apparently a Cantor set͒. See Appendix A for further description of Quadwebs. models from mediocre ones. For this reason, we assert that today, as this central step of modeling moves across the sci- quantifying the quality of a model is an essential problem in ences, we in the applied dynamical systems community still science. A primary interest of this work will be to develop tend to choose the “best modelЉ from a model class in an principles and methods to compare dynamical systems when intuitive manner. Our research here intends to address this they are not necessarily equivalent ͑in the sense of conju- issue of quantifying the quality in a mathematically grounded gacy͒, but in a manner that respects the behavioral implica- manner. tions of conjugacy. A standard approach to quantifying model accuracy is to A. Comparing dynamical systems: On conjugacy measure how “close” the model is to the original system, and nonhomeomorphic commuters where “close” depends on what aspects of the system we are → → trying to model. In many cases, prediction is our modeling Given two dynamical systems, g1 :X X, and g2 :Y Y, goal, such as when forecasting the weather. The quality of the fundamental departure from a typical measurement of prediction is grounded in standard numerical analysis on Ba- approximation between the two dynamical systems is that we nach spaces: the “goodness” of short-term predictions is do not directly compare g1 and g2 under an embedding in a ͑ ʈ ʈ ͒ based on measurement of residual error. However, in dy- Banach space e.g., measuring, say, g1 −g2 L2 because such namical systems, a model’s quality is typically not based on measurements pay no regard to the central equivalence rela- such error analysis. Continuing, for the sake of example, in tionship in dynamical systems, namely conjugacy. Two sys- → the field of meteorology, we cite a famous historical example tems are conjugate if there is a homeomorphism h:X Y ͑ to highlight this long recognized issue. Consider Lorenz’s between the underlying phase space h must be 1-1, onto, −1 ͒ 1965 paper about his 28-variable ODE model of the continuous, and h must be continuous , and h must com- 10 ෈ weather. The model consists of Galerkin’s projection of a mute the mappings at each point x X. This is not only the two-level geostrophic model for fluid flow in the atmosphere. fundamental equivalence relationship of the field, it reflects About the matter of choice of parameter values that are ini- the practical notion that h is an “exact” change of coordi- tially free in the model, the tuning of which led to dramati- nates so that the mappings behave exactly the same in either cally different dynamic behavior due to a plethora of pos- coordinate system. See in Fig. 1 our new quadweb represen- sible bifurcations, Lorenz says, “Our first choice of constants tation of a commuter and Appendix A for further description. → lead to periodic variations. Subsequent choices yielded ir- We give the name “commuter” to any function f :X Y regular….” He relied on his expert knowledge as a meteo- that satisfies the commuting relationship, rologist concerning what “reminded” him of oscillations of f ؠ g = g ؠ f , ͑1͒ realistic weather. Even by the time of his 1998 paper on a 1 2 40-variable model,11 we quote, “Regardless of how well or and note that a commuter will be a conjugacy only if it is a how poorly the equation of the model resemble those of the homeomorphism. The commuter provides a matching be- atmosphere, it is essential to know, before proceeding with tween trajectories for g1 and g2; over- and/or under- our experiments, how closely the model resembles the atmo- representations are reflected as 1-1 and onto problems in f, sphere in its behavior.” Therefore, we summarize that even while trajectories that permit matching only for finite time

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FIG. 2. ͑Color͒ Modeling a noisy logistic map. ͑Left͒ A “noisy” logistic FIG. 3. ͑Color͒͑Left͒ Logistic map with “just a little” extra dynamics. ͑ ͒ ͑ ͒ map g1 blue is obviously near the full logistic map 4x 1−x ,inbothany These maps are not as far from conjugacy as appearances might indicate, p͓͑ ͔͒ ജ 1͓͑ ͔͒ p r L 0,1 , p 1, as well as in the sup-norm, but not in the C 0,1 norm, and they are obviously not close in any L or C norm. That g1 has two ͉ Ј͑ ͉͒ ⌺ ⌺ as the map was constructed with slope g1 x =10 everywhere that it exists. humps, and g2 has one hump, which suggests that subshifts of 4 and 2, ͑ ͒ The green circles graph a tent map g2 with vertical extension which is our respectively, are required to represent the dynamics. In fact, g1 behaves as if “best” model within the family of symmetric tent maps. ͑Right͒ The result- it is almost conjugate to a trapezoid map ͑Ref. 37͒, also known as a gap-map ing commuter f between g1 and this tent map g2 is reminiscent of the ͑Ref. 38͒, since it does not take a large perturbation to replace the extra homeomorphism h͑x͒=1/2͓1−cos͑␲x͔͒ between the full logistic map and humps of x෈͑0.25,0.5͒, with a horizontal line segment. ͑Right͒ Note the the full tent map. Whereas h is a diffeomorphism, f is not even a homeo- nonmonotonicity in the commuter, indicating that g1 has dynamics that are morphism, though our visual impression is that we would not need to move not matched by g2. In the sense of defect, the tent map shown is the best the points ͑on the graph of f͒ very far to achieve a homeomorphism. dynamical match to the blue map; even though we can see clearly that there are closer tent maps in an L2 sense, they are not dynamically closer. are related to discontinuities in f. Our fundamental approach is that we suggest to develop measures of commuters f to Example Problem: Logistic map with “just a little” extra quantify “how much” the f may fail to be a homeomorphism. dynamics. In Fig. 3, we see a map with two humps, which is Such measures quantify the dissimilarity of g1 and g2 in a clearly not conjugate to any one hump map such as the tent manner that respects the fundamental philosophy of dynami- map. However, in some sense, the dynamics are “mostly” cal systems. very similar to those of the tent map. Inspection of the orbit equivalence through the commuter function in Fig. 3 ͑right͒ B. Examples of commuters of “close” dynamical clearly shows the failure of f to be a homeomorphism due to systems the failure of all four of the properties of continuity both ways, one-one-ness, and onto-ness. In a measured sense that Our primary interest is in comparing dynamical systems we call “homeomorphic defect” defined in Sec. V, the failure with methods that respect the notion of conjugacy. We will is not large. We will conclude that a large fraction of the use the commuter to indicate how well the systems are dynamics of g “mostly” match most of the dynamics of g , matched ͑from a dynamical perspective͒ by observing the 1 2 suggesting that the model with one hump is reasonable. deviation from homeomorphism. These examples are indica- More details of this example are in the figure caption to tive goal problems, selected to allow for a clear presentation Fig. 3. in the one-dimensional setting. The eventual goal of our work will be to extend this approach to compare multivariate C. Some background systems, including diffeomorphisms and flows. Although the ͑ ͒ examples and constructions of this paper focuses on one- In the above examples, the “models” g2’s were chosen dimensional examples, the measure theoretic definitions of such that the commuter had a small homeomorphic defect. defect are meant to generalize naturally. An alternative, traditional method is to consider relative en- Example Problem: Modeling a noisy logistic map.Itis tropies between the systems. For the noisy logistic map in ͑ ͒ ͑ ͒ well known that the logistic map g1 x =rx 1−x is conjugate Fig. 2 ͑which is semiconjugate to a shift map embedded in a ͑ ͒ ͑ ͉ / ͉͒ ͒ to the tent map g2 x =a 1−2 x−1 2 for parameter values symbol space of many symbols , selecting a model to mini- r=4 and a=2. The conjugacy h͑x͒=1/2͓1−cos͑␲x͔͒ pro- mize the entropy difference would mean that we would vides perhaps the most studied example in the pedagogy of choose a very steep and high tent map. In an experimental dynamical systems for teaching conjugacy. If r is perturbed situation, depending on our ability to resolve the spatial fine even slightly, the maps fails to be a conjugate. Now consider scales of the thin humps, the noisy map will appear more like ͑ ͒ a system with a much stronger perturbation, like a noisy the original logistic map of hT =ln 2 than like a map of version of the logistic map, which is obviously not conjugate higher entropy, and we might prefer that a simplifying model to the tent map. The commuter shown in Fig. 2 ͑right͒ is not should reflect this lower entropy. We sometimes refer to a homeomorphism, since it fails at least one-one-ness, which well-modeled systems as being “almost conjugate.” There is we see immediately upon inspection. The commuter gives a notion of almost conjugacy already in the symbolic dynam- the orbit equivalence between the two maps and is the key to ics literature,7,12 through “orbit equivalence,” used to com- understanding the quality of a model. One could argue that pare two shift spaces, which relies on the existence of a the tent map shown is a good candidate to model this “noisy factor map between the two shift spaces. Although the term logistic map” because although it simplifies the small-scale “defect” has been used to describe the degree to which such dynamics, it captures the main features of the large-scale a factor map fails to be injective,13 we define homeomorphic orbit structure. We quantify this argument by measuring the defect using an entirely different construct for use in a very defect of f using techniques described in Sec. V. different setting—where systems are not equivalent. Ergodic

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theory provides other metrics on equivalence classes of dy- about x=1/2. We denote this subfamily as T, where TʚS. ෈T namical systems, based on notions such as Kakutani equiva- An arbitrary member T␯ is defined by T␯ ªS1/2,␯. lence or metric isomorphism.14 Our work is complementary First, we note the following lemma regarding the exis- to that work, but it is somewhat different in that we can tence of a conjugacy: ͑ S directly compare two maps even when they are not shift Lemma 1. Let Sa,b be a particular member of . Then maps͒ and explicitly construct the commuter function—the there exists a ␯ such that S is conjugate to T␯ ෈T. 0 a,b 0 realization of the factor map. Our goal is to provide a mea- Sketch of Proof: Kneading sequences need to be ␯ sure between equivalence classes based on topological con- matched, and we know that there is a value of 0 that jugacy. Quantifying the similarity between two dynamical matches the kneading sequences and hence symbolic dynam- systems directly through measuring a change of coordinates ics by a so-called intermediate value theorem of kneading between them is a departure from general literature, and the sequences, which can be found in either Misiurewicz and only paper we find remotely in the same vein is that of Yuan– Visinescu20 or Collet and Eckmann.21 Although matching Yorke et al.,15 whereas the methods and scope were quite symbolic dynamics is not enough to prove that the maps are different. We believe the idea is natural and general. conjugate, it is necessary. Similarly, the fact that both maps are strictly monotone is enough to guarantee that monotone laps map to monotone laps, from which it can be shown that II. ON CONSTRUCTION OF COMMUTERS there is a match between any two points with the same sym- In this section, we use the tent map as an easy setting to bolic itineraries. The match of symbolic dynamics does guar- introduce how the commuter, or would be conjugacy, can be antee a match between the eventually periodic points. Be- explicitly constructed by a fixed-point iteration scheme. In cause both maps are assumed to be everywhere expanding, this simple setting, we will rigorously prove several aspects the matching can be extended to the entire interval, with h of the existence and convergence of the fixed-point iteration. and h−1 continuous. ᮀ In subsequent sections, we will see that for more compli- The remainder of this section is constructed under the cated problems, the methods seem to work numerically well presumption that we have chosen a particular value for a and beyond our sufficient theorems. In subsequent sections, we b, and though they are arbitrary, they remain fixed. The map ϵ will introduce the commuter’s role in measuring homeomor- S Sa,b will be used to symbolize this arbitrary but fixed phic defect. map. To find the conjugacy whose existence is indicated by While here we propose to use commuters between dy- Lemma 1, we need to solve the functional equation namical systems in a unique way as part of developing our ͒ ͑ ͒ ͑ ؠ ͑ ͒ ؠ defect measure, for the construction step, we should point to S h x = h T x . 3 16 the line of work dating to Schub in 1969, and further stud- In particular, one must find an h that not only satisfies Eq. ͑3͒ ied, for example, in Ref. 17 and reviewed in the popular but is also a homeomorphism. In general, there is no direct 18 textbook by de Melo and van Strien. These are related to technique to find such an h. Instead, we propose an indirect our Lemmas 1–3. In particular, see Theorem II 2.1 in Ref. solution approach: we create a fixed-point iteration scheme 18. Therein, endomorphisms, including general conjugacies, that converges to a solution. The rest of this section describes are constructed using methods based on the contraction map- the development of that iterative scheme, which we decom- ping principle related to what we describe in Sec. II. Gener- pose into three components: alization for the multivariate expanding case can be found in Ref. 16. A major difference between our work and that of • Creating a contraction mapping that generates solutions to Shub is our interest in comparing arbitrary domains between the commutative diagram. the maps, which has required that we develop an extended • Explaining why the result of that contraction yields a con- ͑ inverse gˆ −1 in Eq. ͑30͒ when defining our contraction opera- jugacy h when S and T are conjugate or, equivalently, 2i ␯ ␯ ␯ ͒ tors, Cg2. when = 0, with 0 known . g1 ␯ • Describing an iterative technique to find 0 when a and b are given, but the required conjugacy parameter value A. A conjugacy between tent maps ͑from Lemma 1͒ is not known. Consider the family of skew tent maps S͑x͒ defined on ͓0, 1͔, with the following restrictions: B. A contraction mapping from the commutative • S͑0͒=0 and S͑1͒=0. relationship • The peak of the tent occurs at S͑␣͒=␤, with 0Ͻ␣Ͻ1. On the interval ͓0,1/2͔, the explicit form for the sym- • To ensure that the map is locally expanding, we require ͑ ͒ ␯ ͑ ͒ metric tent map is T x =2 0x. Substituting into Eq. 3 gives ͒ ͑ ͒ max͑␣,1 − ␣͒ Ͻ ␤ ഛ 1. ͑2͒ ؠ ͑ ͒ ͑ ␯ S h x = h 2 0x . 4 Define this family S, where the family is parametrized To write an explicit description of S, we note that the interval by ␣ and ␤, the coordinates of the peak of the tent. Denote a ͓0,1/2͔ describes the domain of the left part of the tent map specific member of this family as S␣,␤, where the coordinates T. Because we are identifying the conjugacy between these of the peak of that tent are at ͑␣,␤͒. Within this family of two maps ͑which preserves the kneading sequence͒, we con- skew tents, consider the subset of maps that are symmetric clude that h͓0,1/2͔=͓0,a͔ because ͓0,a͔ is the domain of

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a f͑2␯x͒ 0 ഛ x ഛ 1/2 b M␯f͑x͒ ͑13͒ ª Ά 1−a · 1− f͑2␯͑1−x͒͒ 1/2 Ͻ x ഛ 1. b

FIG. 4. ͑Color online͒ Satisfying the functional equation. Equation ͑7͒ can The constraints on the parameters a,b and ␯ are required to be viewed as a process: take h͑x͒͑panel 1͒ and make a copy, shrunken by ensure that F is mapped into itself, but they also cause the / ␯ ͑ ͒ a b in the vertical and by 2 0 in the horizontal panel 2 . Take a second ͑ ͒/ operator to be a contraction. copy, scaled the same horizontally, but vertically scaled by 1−a b. Rotate F this copy by 180 degrees and place it in the upper right portion of the unit Lemma 2. M␯ is a uniform contraction on , where the square ͑panel 3͒. Then truncate the left copy to the interval ͓0,1/2͒ and the contraction is with respect to ʈ·ʈϱ. right copy to ͓1/2,1͔. The result ͑panel 4͒ should return the original h͑x͒. Proof. Define a 1−a ␭ = maxͩ , ͪ. ͑14͒ the left part of the S map. On that interval, S͑u͒=b/au, and b b ͑ ͒ substitution into Eq. 4 gives Then 0Ͻ␭Ͻ1. We compute ʈ ʈ ͉ ͑ ͒ ͑ ͉͒ ͑ ͒ b M␯f1 − M␯f2 ϱ = sup M␯f1 x − M␯f2 x . 15 h͑x͒ = h͑2␯ x͒. ͑5͒ x a 0 We decompose this problem into the two cases xഛ1/2 and Similar logic applied to the interval ͑1/2,1͔ gives xϾ1/2. For the first case, ͉ ͑ ͒ ͑ ͉͒ b sup M␯f1 x − M␯f2 x ͑1−h͑x͒͒ = h͑2␯ ͑1−x͒͒. ͑6͒ ഛ ഛ / 1−a 0 0 x 1 2 a ͑ ͒ ͯ ͑ ͑ ␯ ͒ ͑ ␯ ͒͒ͯ Therefore, the conjugacy function h x must satisfy the func- = sup f1 2 x − f2 2 x tional equation, 0ഛxഛ1/2 b a ഛ ͉ ͑ ͒ ͑ ͉͒ ഛ␭ʈ ʈ a sup f1 y − f2 y f1 − f2 . h͑2␯ x͒ 0 ഛ x ഛ 1/2 b 0ഛyഛ1 b 0 h͑x͒ = ͑7͒ Ά 1−a · Similarly ͑ ␯ ͑ ͒͒ / Ͻ ഛ 1− h 2 0 1−x 1 2 x 1. ͉ ͑ ͒ ͑ ͉͒ b sup M␯f1 x − M␯f2 x 1/2Ͻxഛ1 ͑see Fig 4 for a graphical representation of this relationship͒. 1−a Since a conjugacy should map turning points to turning ͯ ͑ ͑ ␯ ͒ ͑ ␯ ͒͒ͯ = sup f1 2 x − f2 2 x points, we note that 1/2Ͻxഛ1 b 1−a ͑ / ͒ ͑ ͒ ഛ ͉ ͑ ͒ ͑ ͉͒ ഛ␭ʈ ʈ h 1 2 = a. 8 sup f1 y − f2 y f1 − f2 . b 0ഛyഛ1 Additionally, evaluating Eq. ͑7͒ at x=1/2, we have So M␯ is a contraction, with contraction constant ␭. Because a ␭ does not depend on ␯, the contraction is uniform. ᮀ h͑1/2͒ = a = h͑␯ ͒, ͑9͒ b 0 With this groundwork in place, we establish the existence of a fixed point of the operator. yielding that Lemma 3. There is a unique f␯ ෈F satisfying

͑␯ ͒ ͑ ͒ M␯f␯ = f␯. ͑16͒ h 0 = b. 10 ෈F Using Eq. ͑7͒ as a guide, we now create an operator Moreover, for an arbitrary f0 , if we define the sequence whose fixed point will satisfy the commutative diagram. We of functions consider the space B͓͑0,1͔,R͒, the set of all bounded func- ͑ ͒ fn+1 = M␯fn, 17 tions from ͓0, 1͔ to the real numbers, which is a Banach space, with norm given by this sequence will converge to f␯: ͑ ͒ ʈ ʈϵʈ ʈ ͉ ͑ ͉͒ ͑ ͒ f␯ ª lim fn. 18 f f ϱ ª sup f x . 11 n→ϱ x෈͓0,1͔ Proof. The lemma is a direct application of the Banach- Then we form the closed subset FʚB͓͑0,1͔, ͒, defined by R Caccioppoli Contraction Mapping Principle,22 and we simply F = ͕f͉f:͓0,1͔ → ͓0,1͔͖, ͑12͒ need to verify that we satisfy the hypothesis of the theorem. Since B͓͑0,1͔,R͒ is a Banach space, with F a closed subset, the set of functions from ͓0, 1͔ to ͓0, 1͔. Then given ͑a,b͒ and M␯ :F→F, the theorem applies, and the conclusions are satisfying max͑a,1−a͒ϽbϽ1, we define a one-parameter immediate. ᮀ family of operators M␯ :F→F for 1/2Ͻ␯ഛ1, Remark: For any chosen ␯, by construction, the fixed

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tories do not have a matching trajectory under T. ␯Ͼ␯ In the case 0, the entropy mismatch is in the other direction, with

h ͑T␯͒ Ͼ h ͑T␯ ͒ = h ͑S͒. ͑21͒ T T 0 T There are now orbits in T that cannot be represented in S. Consequently, the commuting function f␯ must “remove” an ͑ ͒ ␯ appropriate amount of entropy from the T dynamics by map- FIG. 5. Color Resolving 0. Fixing a=0.25 and b=0.9, we graph f␯ for ␯=0.72 ͑red͒ and ␯=0.73 ͑blue͒. The zoomed panel ͑right͒ illustrates that ping multiple points in the domain to single points in the ␭ˆ ͑0.72͒Ͻ0 ͑the red curve jumps up at x=0.5͒ while ␭ˆ ͑0.73͒Ͼ0 ͑the blue range, so f␯ is not 1-1. Numerically, we observe similar dis- curve jumps down͒. continuities at x=1/2 ͑repeated in accordance with the kneading sequence͒. We use the same measuring function ␭ˆ ͑␯͒ as defined in Eq. ͑19͒, but we note that ␭ˆ ͑␯͒Ͼ0 when point f␯ will satisfy the requirements of the commutative ␯Ͼ␯ . diagram, at least over some portion of its domain ͑see Ap- 0 Despite any limitations of numerical approximations of pendix A for more detail͒. The question remains as to ␭ˆ ͑␯͒ whether it is a conjugacy function between the two maps S , we know analytically that and T␯. If the two maps are conjugate, which we know is ␭ˆ ͑␯ ͒ ͑ ͒ possible by Lemma 1 by choosing the parameter to be ␯ 0 =0. 22 =␯ , then it is straightforward to check that the f␯ produced 0 0 Consequently, we can apply typical scalar root finding algo- by the fixed-point iteration scheme is that conjugacy func- rithms ͑bisection, or secant algorithms, for example͒ to ap- tion, and we call it h͑x͒= f␯ ͑x͒. It is important to note for ␯ 0 proximate 0. much of the following development that even if the two maps S and T␯ are not conjugate and so no conjugacy func- III. CONSTRUCTING THE CONTRACTION MAPPING tion h͑x͒ exists, the fixed-point iteration still produces a FOR GENERAL TRANSFORMATIONS function f␯ satisfying the commutative diagram. However, it cannot be a homeomorphism. In this section, we consider more general classes of dynamical systems than the tent maps examined in Sec. II A. ␯ Our goal is to develop a methodology that will allow us to C. The search for 0: A problem of parameter estimation “solve” the commutative diagram. Our primary tool will be the construction of an appropriate operator whose fixed point Suppose we are given a chaotic map from the family S. yields a solution to the commutative diagram. We call this ͑In other words, ͑a,b͒ are fixed.͒ By Lemma 1, we know that operator the commutation operator. Our approach to defining there is a ␯ such that map T␯ is conjugate to S. In this 0 0 the operator will be to generalize the technique of Sec. II A, section, we give a short explanation of how we find that which constructed a contraction operator to solve the appro- conjugate map. priate functional equation. We explain our method in two ␯␯ When 0, the maps Sa,b and T␯ are not conjugate, and parts, first by showing the construction process when the while the existence of f␯ as the fixed point of a contraction is dynamical systems are conjugate, and then we show how to guaranteed by Lemma 3, f␯ fails to be a homeomorphism. adapt that method to the case in which the maps are not ␯Ͻ␯ We find that for 0, f␯ is monotone increasing, but not conjugate. For this general case, we do not prove that the continuous and not onto ͓0, 1͔. The largest “gap” is at x commutation operator is a contraction, but on all numerical =1/2. Because f␯ satisfies a recursive formulation ͑which examples we have tested, it has converged to an approximate creates a self-similar structure͒, a scaled version of this de- solution. fect appears at all x coordinates associated with the kneading sequence of T. We define A. Constructing the commutation operator for conjugate maps ␭ˆ ͑␯͒ ͑ ͒ ͑ ͒ ª lim f␯ x − a 19 ؠ ؠ ͒ ͑ −͒ / ͑→ x 1 2 Here we must solve the commuting Eq. 1 , f g1 =g2 f, for a general one-dimensional g and g . To phrase this equa- ␭ˆ ͑␯͒Ͻ 1 2 as a way of measuring this “signed defect,” with 0 tion as a fixed-point problem, we would like to solve for f on ␯Ͻ␯ ͑ ͒ when 0 see Fig 5 . We discuss measuring homeomor- the right-hand side, but because we are interested in the case phism defects and easier to compute suitable surrogates in in which g2 generates chaotic dynamics, it will not be invert- Sec. V. ible. As we did with the tent maps, we tackle this problem by As a short explanation for the gaps, we note that since providing a piecewise definition for the operator, where on −1 h ͑T␯͒ Ͻ h ͑T␯ ͒ = h ͑S ͒, ͑20͒ each piece we use a restricted subset of Y to define the g2 . T T 0 T a,b −1 Note that an extended version of g2 must be defined, which −1 the only way that we match equivalent orbits under the T and we will call gˆ 2 in order to well define the contraction. S dynamics is to restrict the domain of S. In other words, In this subsection, we assume that g1 and g2 are conju- there must be orbits of S that cannot be represented by T, and gate. Therefore, there is a homeomorphism h that satisfies the “gaps” indicate those initial conditions in S whose trajec- the commutative diagram, and this h is called a conjugacy. It

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−1 is our goal to have h be a fixed point of our constructed requirements, as stated, do not uniquely define gˆ 2i . We have operator. We demand that h map monotone segments of the chosen to retain this flexibility because we find that a judi- graph of g1 onto monotone segments of g2. To simplify the cious choice of extension may ease the numerical implemen- explanation, we assume that X and Y are compact intervals. tation and provide additional insight into the relationship be- ͕ ͖ ϵ͓ ͔ Let x= x0 ,x1 ,...,xn , with X x0 ,xn , and g1 monotone on tween the systems. In this paper, the examples have used one ͓ ͔ ͑ ͒ −1 each interval xi−1,xi , for i=1,...,n. This notation results in or the other of the following strategies: i assume gˆ 2i is ͓ ͔ ͑ ͒ a natural decomposition of X into a union of disjoint sub- constant on Y −g2 IYi ,or ii when g2 is piecewise linear, −1 −1 intervals, then simply assume that gˆ 2i lies on the same line as g2i . Proposition 1. Let g and g be conjugate, with h:X I = ͓x ,x ͔, ͑23͒ 1 2 X1 0 1 →Y the conjugacy. Then h is a fixed point of the commutation operator, Cg2. I = ͑x ,x ͔, i =2, ...,n, ͑24͒ g1 Xi i−1 i Proof. Take an arbitrary x. Then there is an i such that ෈ ഫ ͑ ͒ x IXi. Because h is a conjugacy, it must map monotone X = IXi, 25 ͓ ͔ i intervals to monotone intervals. In particular, h IXi =IYi, which implies that y h͑x͒෈I . We apply the definition of ͑ ͒ ª Yi where we have arbitrarily chosen the closure conditions of the commutation operator, the intervals. Because of the conjugacy of the dynamical ϵ͓ ͔ Cg2h͑x͒ = gˆ −1 ؠ h ؠ g ͑x͒. ͑31͒ ͖ ͕ systems, there must exist y= y0 ,...,yn , with Y y0 ,yn g1 2i 1 ͓ ͔ and g2 monotone on each interval yi−1,yi , with a similar -Because h is the conjugacy, we know that hؠg =g ؠh. Sub 23 decomposition of Y into a union of disjoint subintervals I . 1 2 Yi stituting into the above equation gives To allow for well defined inverse functions, we denote ͒ ͑ ͒ ͑͒ g2 ͑ ͒ −1 ؠ ؠ ͑ ͒ ͑ −1 ؠ C h x = gˆ g2 h x = gˆ g2 y . 32 g = ͉g ͉ . ͑26͒ g1 2i 2i 2i 2 IYi ͉͒ ෈ ͉͑ˆ −1ؠ Because f is a conjugacy, it must map monotone inter- Because y IYi and g2i g2 IYi is equivalent to the identity, vals of X to monotone intervals of Y, such that we may write we have g2 ͑ ͒ ͑ ͒ ͑ ͒ ͓ ͔ ͑ ͒ Cg h x = y = h x . 33 f IXi = IYi, i =1, ...,n, 27 1 by which we infer the identity ᮀ Although h is a fixed point of the operator, we have not ͒ ͑ ͔ ͓ ؠ ؠ −1 ͔ ͓ f IXi = g2i g2 f IXi . 28 established that we can use an iterative scheme to find h.In We can now rewrite the commuting equation ͑1͒ as a system the best situation, we would like the commutation operator to ͑ of equations, be a contraction as it was for the class of tent maps considered in Sec. II A͒. In application to maps with chaotic attrac- ͒ ͑ ͔ ͓ ͔ ͓ ؠ ؠ −1 g2i f g1 IXi = f IXi ,i =1, ...,n. 29 tors, we have found that iteration under the operator numeri- Having now solved for f on the right-hand side ͑RHS͒ of Eq. cally converges to a fixed-point function. ͑29͒, we can use this formulation to construct the commuta- B. The commutation operator for maps tion operator. Let F be the set of functions from X to Y. Then that are not necessarily conjugate we define the operator Cg2 :F→F, which takes F෈F to „ … g1 Cg2F by We now consider the problem of creating a commutation g1 operator when map g1 and g2 are not necessarily conjugate. ͒ ͑ g2 ͑ ͒ −1 ؠ ؠ ͑ ͒ ෈ C F x = gˆ F g x x IXi. 30 g1 2i 1 We would like this operator to be a contraction, so that its fixed point could be found by fixed-point iteration. More- ͑ ͒ Note the slight change in notation between Eqs. 29 and over, we want the fixed point to satisfy the commutative ͑ ͒ ˆ −1 −1 30 in that we now use g2i instead of simply g2i , where we diagram. In this section, we will simply address the problem still need to define and explain the former. We remark that of creating the operator such that the fixed point of the op- −1 ͓ ͔ g2i is defined only for the interval g2 IYi , which may not be erator satisfies the commutation diagram on some positive ͒ ͑ ؠ all of Y. However, F g1 x may be any value in Y. Conse- measure subset. The primary difference between this prob- −1 ˆ −1 quently, we need to extend g2i to all of Y. We define g2i such lem and that of the preceding section is that because the that it satisfies the following: systems are not conjugate, there is no a priori reason that we −1 should be able to match monotonic intervals of one dynami- • gˆ 2i is continuous on Y. −1ϵ −1 ͓ ͔ cal system to monotonic intervals of the other. Consequently, • gˆ 2i g2i on g2 IYi . −1 ͓ ͔ the method requires some ad hoc component that specifies • gˆ 2i is Lipshitz continuous on Y −g2 IYi , with Lipshitz constant LϽ1. this matching. As in the preceding section, the underlying principle is that we will construct an operator whose fixed Simply by expanding the domain of definition for each of the point satisfies the commutative diagram. If the resultant op- inverses, we ensure that for each x෈X, Cg2F͑x͒ exists, so erator is a contraction, then the fixed point can be found by g1 that the commutation operator is well defined. The require- an iterative method. In general, g1 and g2 are not conjugate, ment on the Lipschitz constant is meant to increase the like- which means that the fixed point is not a conjugacy. We will lihood that the operator may be a contraction. However, the use commuter to denote a fixed point of this general commu-

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jsp 013118-8 J. D. Skufca and E. M. Bollt Chaos 18, 013118 ͑2008͒ tation operator, where the term appropriately describes that that in some modeling situations there will be good reasons the fixed point satisfies the commutative diagram. to take alternative approaches to those outlined below. To motivate our methodology, we start with the “idea” • Choose n to be as small as practical. As n increases, the that the commuter, f :X→Y should act like a change of vari- modeler is required to make more choices regarding which ables, associating a trajectory of dynamical system g with a 1 intervals of X should be matched with which intervals of trajectory of g . In mapping an interval I ʚX to a f͓I ͔ʚY, 2 x x Y. Unless the modeler has an a priori reason to force a we would really like to be able to associate trajectories of particular matching, it is better to allow the algorithm to points in I to trajectories of points in f͓I ͔. If the systems are x x find a matching. Consequently, it is generally counterpro- not conjugate, then the “matching” will not be perfect. When ductive to choose n to be larger than n , where n is the the two systems are conjugate, the commuter is continuous, 2 2 number of monotone sequences of g . On the other hand, mapping intervals to intervals, and we are able to easily iden- 2 g must be monotone on each interval I to allow the tify an appropriate matching by the maximum intervals on 2 Yi inverse to be defined. So if the graphs of gˆ −1 are to cover24 which the dynamical systems are monotone. In essence, it is 2i the graph of g , we will need at least n different inverse this matching that prescribes the operator. 2 2 functions, requiring nജn . Therefore, we typically choose Construction of the commutation operator requires that 2 n=n . the mathematician, now acting as a modeler, make some 2 • We usually require IϵX. As a slight weakening of that choices regarding the particular matching scheme to be em- condition, we might simply require that I be a closed in- ployed. To describe our technique, we will decompose the terval in X. The basic idea is that we would like the com- problem into two components: first, we define a set of mini- muter to map X to Y, so we need to cover as much of X as mal requirements to constructing the operator; secondly, we is possible. define a set of recommended choices for the construction, •Ifnജn , we will construct gˆ −1 so that I are disjoint and where we provide some justification for those recommenda- 2 2i Yi that ഫ I is a closed interval. Typically, ഫ I ϵY. This tions. i Yi i Yi construction allows the covering of g described above. The following are requirements for construction of a 2 Additionally, because the IXi have been ordered, we as- commutation operator: −1 sume that gˆ 2i have been chosen to reflect a similar order- 1. Choose an integer n, which will be the number of inter- ing on the IYi. vals for which we will prescribe a matching. ͕ ͖n Remark: For the divergence measurements of the com- 2. Choose a collection of disjoint subintervals IXi i=1, ʚ ഫ mutator f discussed in the next section, we would like to where IXi X, and I= iIXi is a closed set that is forward have at least existence and uniqueness of f in all of the invariant under g1. Without loss of generality, we as- general settings just discussed. Our experience indicates that sume that these intervals are ordered, such that if x1 ෈ ෈ Ͻ Ͻ a useful f does exist uniquely for all of the widely varied IXi and x2 IXj, then x1 x2 whenever i j. 3. For each i෈͕1,...,n͖, assign an inverse function modeling choices of domains, partitions, and nonconjugate −1 → systems we have made. For the time being, however, the gˆ 2i :Y Y which satisfies the following: generality of our method remains unproven, although our −1 • gˆ 2i is continuous on Y. remarks are suggestive of how the methods of proof de- ʚ −1 • There is an associated interval IYi Y, such that gˆ 2i acts signed for specific examples may generalize. Observe that like an inverse to g on the interval I . Equivalently, for 2 Yi the expansiveness of g2 is needed to give the contraction of ෈ −1͑ ͑ ͒͒ −1 −1 all y IYi,gˆ 2i g2 y =y. g and consequently its extension gˆ to prove a contraction −1 ͓ ͔ 2 2 • gˆ 2i is Lipshitz continuous on Y −g2 IYi , with Lipshitz mapping converging to f as shown in Lemmas 1–3. constant LϽ1.

Using the above minimal choices, the definition of commu- IV. MEASURE OF MOSTLY CONJUGATE ͑ ͒ tation operator given in Eq. 30 remains notationally cor- Suppose we have two dynamical systems, rect. However, we need to recognize that the construction ͕ ͖ ͕ −1͖ → → ͑ ͒ depended upon the actual choices for IXi and gˆ 2i , and we g1:X X, g2:Y Y. 35 should formally treat them as parameters in describing how the operator acts on functions, When the two systems are topologically conjugate, then the dynamics of one system completely describe the dynamics of g2 ͑ ͒ϵ g2͕͑ ͖ ͕ −1͖͒ ͑ ͒ −1 ؠ ؠ ͑ ͒ ෈ C F x C IXi , gˆ F x gˆ F g x , x IXi. the other. However, if they are not conjugate, we may find g1 g1 2i ª 2i 1 that “some” of the dynamics of one system can be described ͑34͒ by the other. We might heuristically judge that “most” of the Based on our experience, we provide the following list dynamics are well represented by the other system. In some of recommendations for formulating the commutation opera- sense, we could consider the two systems to be “close,” but tor, where our goal is to improve the utility of the resultant that description is only reasonable if we can describe a “dis- fixed point. In particular, if the systems are conjugate, we tance from conjugacy.” As an application, we consider this D would like the fixed point to be the required conjugacy. prototypical problem: Fix g1, and consider some family of When the systems are not conjugate, we still would like it to dynamical systems; find the element g*෈D that most act as a commuter from X to Y, while retaining as much of closely approximates the dynamics of g1. Topological conju- the character of a homeomorphism as possible. We recognize gacy defines two systems as having the same dynamics, and

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jsp 013118-9 Homeomorphic defect of modeling Chaos 18, 013118 ͑2008͒ any notion of “distance to conjugacy” ought to provide a onto the family G. We denote ¯g, the projection of g into G,by means of determining the extent to which the dynamics are ͑ ͒ ͑ ͒ similar. In this section, we propose a construct that allows ¯g = arg min d g,˜g . 38 ˜g෈G such a measurement. In Sec. III, we described a technique for finding a com- From a theoretical aspect, we need to be concerned with both muter, f, from g1 to g2. Although f satisfies the commutative the existence and uniqueness of this minimizer. In this paper, diagram, it need not be a homeomorphism ͑and, therefore, we will simply argue that our approach is meant to give a not a conjugacy͒. Let ␭͑f͒ be a measure of how far f deviates computable approximation, and within that framework of nu- ␭͑ ͒ϵ from a homeomorphism, with f 0 when g1 and g2 are merical implementation, it appears that our projection opera- conjugate. In Sec. V, we give a more thorough description of tion is sufficiently robust. Using this projection, we now de- ␭, but for now, we simply state that the measuring function ␭ fine a new measurement function DG by should be defined with sufficient flexibility to measure f rela- ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ tive to the “important” subset of X and Y, where “important” DG g1,g2 = d g1,g1 + d g1,g2 + d g2,g2 . 39 and the associated measure would be problem-dependent and We make the following remarks regarding DG: according to the modeler’s choice of what to compare. If we ෈G ⇒ ͑ ͒ define CF͑g ,g ͒ to be the set of all commuters from g to •Ifg1 , then we may choose g1 =g1 d g1 ,g1 =0. 1 2 1 ෈G ⇒ ͑ ͒ g , then for a given ␭, we can define •Ifg2 , then we may choose g2 =g2 d g2 ,g2 =0. 2 ෈G ͑ ͒ϵ ͑ ͒ •Ifg1, g2 , then DG g1 ,g2 d g1 ,g2 . ␦͑ ͒ϵ ␭͑ ͒ ͑ ͒ G g1,g2 inf f . 36 •If is a sufficiently large family such that g1 and g2 can be ෈͑ ͒ f g1,g2 ͑arbitrarily͒ well approximated, then we may expect As an expository description of ␦, we may think of a com- ͑ ͒ ͑⑀͒ ͑ ͒ ͑⑀͒ d g1,g1 = O ,d g2,g2 = O muting function f as taking us from dynamical system g1 to ␭͑ ͒ ␦ g2, and f measures a cost. Then would describe a great- and est lower bound for that cost. In general, we would not re- 25 ␦͑ ͒ ͑ ͒ ͑ ͒ ͑⑀͒ ͑ ͒ quire that the cost function be symmetric, so that g1 ,g2 DG g1,g2 = d g1,g1 + O . 40 ␦͑ ͒ need not be the same as g2 ,g1 . However, we will require ␦ We also recognize that an alternative approach in choosing that =0 if g1 and g2 are conjugate. Although ␦ appears to be a useful theoretical construct, projections g1 and g2 would be to choose a projection pair ͑ ͒ ͑ ͒ ͑ ͒ we have no method for performing the optimization to evalu- that minimizes the sum, d g1 ,g1 +d g1 ,g2 +d g2 ,g2 . How- ate. Therefore, we also define a less global definition by tak- ever, that approach might add significant computational com- ing the following approach: In Sec. III, we described tech- plexity to the optimization problem, and we have found the niques to construct the commutation operator, C, for a given simpler approach to be sufficient in application. g1 and g2. Suppose that within a certain class of problems, we choose a particular canonical technique to construct the A. Manifesto for our particular choice for G commutation operator such that given a particular choice of Consider the family of dynamical systems T defined as g and g , the operator is uniquely defined. This unique 1 2 follows: choice of operator will have an associated fixed point, fC, which is a commuter. Then we can use this commuter to • Each f ෈T is continuous and piecewise linear on ͓a,b͔. evaluate the cost of transforming between the dynamical sys- • For each f ෈T, ͉TЈ͑x͉͒=c, constant wherever the derivative tems, defining exists. • For each f ෈T, ͉TЈ͑x͉͒ exists for all but perhaps a finite set ͑ ͒ϵ␭͑ ͒ ͑ ͒ d g1,g2 fC . 37 of points. In practical application, given a dynamical system g,we We call T the the set of constant slope maps. In several may wish to restrict the comparison to dynamical systems ˜g applications, we note that choosing GʚT results in some from a particular family G. Some examples of such restricted nice properties for the projections because they inherit these families include the following: properties from dynamical systems characteristics of con- • skew tents stant slope maps. • symmetric tents • constant slope maps V. MEASURING THE DEVIATION • polynomial maps of degree n. FROM HOMEOMORPHISM ͑ ͒ → The point of using these restricted families is that i it may Suppose we have two dynamical systems, g1 :X X and → → be easier to provide a methodology for producing unique g2 :Y Y. Additionally, suppose f :D R is a commuter, f ؠ ؠ ͒ ͑ commutation operators; ii we may choose families that are g1 =g2 f.Iff were a homeomorphism, then g1 and g2 described by a finite set of free parameters, making it easier would be conjugate. However, if we know that the two dy- to search within that family; ͑iii͒ the properties of the canoni- namical systems are not conjugate, then f must fail to be a cal system may be universally understood. When applying homeomorphism. We desire to build a metric that measures this restriction to the general problem of comparing an arbi- the extent to which f fails to be a conjugacy. Our general trary g1 and g2, it is then necessary to project the problem strategy will be to define a homeomorphic defect, which pro-

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vides a weighted average based on measurements of each possible failure. We denote ␭ ͑ ͒ ͕ ͖ O f = amount that f is not onto , ␭ ͑ ͒ ͕ ͖ 1-1 f = amount that f is not 1-1 , ␭ ͑ ͒ ͕ ͖ C f = amount that f is not continuous , ␭ ͑ ͒ ͕ −1 ͖ C−1 f = amount that f is not continuous , −1 ͑ ͒ ␭ where we note that f may not be well defined. While there FIG. 6. Color Illustration of onto deficiency. See the definition of O, Eq. are certainly many ways to define each of these components, ͑46͒. The space X is represented on the horizontal, with Y the vertical. The a specific definition scheme should satisfy the following: sets of interest, D1 and D2, are colored red and green. On each graph, we show an example of the graph of a commuter, f͑x͒. The “onto-deficiency” is 26 ␮ ␭ ͑ ͒ജ computed from a 2 measure of the yellow portion. The right-hand example • O f 0, with equality when f is onto. ␭ ͑ ͒ജ illustrates that ͑1͒ we are only interested in measuring the extent to which • 1-1 f 0, with equality when f is 1-1. D is covered ͑so the lower gap is not a problem͒, and ͑2͒ we are only ␭ ͑ ͒ജ 2 • C f 0, with equality when f is continuous. allowed to use the part of f that is over D1. ␭ ͑ ͒ജ −1 • C−1 f 0, with equality when f is continuous. We define homeomorphic defect of f, denoted ␭͑f͒,asa inherited from the usual topology. Our notation is that → convex combination f :D R is a commutator. Generally, we think of D1 as being ␭͑ ͒ ␣ ␭ ͑ ͒ ␣ ␭ ͑ ͒ ␣ ␭ ͑ ͒ ␣ ␭ ͑ ͒ ͑ ͒ a restriction to a set of interest, but we do not preclude the f = 1 O f + 2 1-1 f + 3 C f + 4 C−1 f , 41 ʚ ͑ ʚ ͒ case that D1 D or similarly D2 R . These situations re- where the weights satisfy flect that sometimes the modeler is interested in a set that is larger than where the commutator is defined. As necessary, ഛ ␣ ഛ ␣ ͑ ͒ 0 i 1, and ͚ i =1. 42 we will simply assume that the subsets resulting from various intersections are ␮ or ␮ measurable whenever this Our decision not to specify a particular choice for the 1 2 measurability is required by some definition. For ease of no- weights allows the flexibility to “tune” this metric for a par- tation, we define ticular application. Then ␮ ͑f͓A͔͒ = ␮ ͑f͓A പ D ͔ പ D ͒͑44͒ ␭͑f͒ ജ 0, with equality when f is a homeomorphism. ͑43͒ 2 2 1 2 for arbitrary set AʚX. The idea is that we want to restrict We note that if the converse were to hold ͑such that 0 defect ourselves to measuring image points that lie in D , whose implied homeomorphism͒, then ␭ could be used to provide a 2 pre-image was in D1. Similarly, we define “distance from conjugacy” for the two dynamical systems g1 ␮ ͑ −1͓ ͔͒ ␮ ͑ −1͓ പ ͔ പ ͒ ͑ ͒ and g2. In this paper, our primary goal is to maintain the 1 f B = 1 f B D2 D1 . 45 flexibility of the definitions ͑to allow broader applicability͒. Indeed, to retain this flexibility, our definitions are measure- B. “Onto” deficiency based and, consequently, we expect that the converse will not hold. To measure the onto deficiency, we desire to measure the fraction of D2 which is not covered by the range of f.We ␭ A. Supporting assumptions and notation define the onto deficiency, O, of the function f by We assume measure space/s ͑D ,A ,␮ ͒ and ␮ ͑f͓D ͔͒ 1 1 1 ␭ ͑ ͒ 2 1 ͑ ͒ ͑ A ␮ ͒ ʚ ʚ A A ␴ O f =1− . 46 D2 , 2 , 2 , where D1 X and D2 Y, 1 and 2 are ␮ ͑D ͒ ␮ ␮ 2 2 algebras and 1 and 2 are measures. D1 and D2 are “chosen” by the modeler, and represent the subsets of X and Y See Fig. 6. that are of interest to the modeler. For example, one might be Because f may have fractal structure with the range of f a Cantor set, it may be difficult to implement Eq. ͑46͒ in interested in comparing the dynamics of g1 and g2 on their forward invariant sets, which might be smaller than the computational practice. As a “suitable surrogate,” we find ␮ whole space of the dynamics. Additionally, by allowing the that if D2 is an interval and 2 is absolutely continuous, we modeler to specify a measure, the “important” parts of the can often quantify the lack of onto-ness by finding the “big- gest hole” in the range of f. Specifically, if we define sets D1 and D2 can be more heavily weighted. In some cases, the dynamics of interest might lie on D =C, with C an in- ͓ ͔ ͑ ͒ 1 G ª D2 − f D1 , 47 variant Cantor set with Lebesgue measure-0. We may still ␮ ␮ ͑ ͒Ͼ then a suitable surrogate is given by choose 1 such that 1 D1 0, which would allow us to measure subsets of D1 in ways that will distinguish there ˜␭ ͑ ͒ ͑ ͒ ͑ ͒ O f ª sup m I , 48 size. Typically, we will be interested in chaotic dynamics, IʚG and assume that X and Y are bounded sets. In all the ex- where I is an interval and m is a Lebesgue measure. Note amples in this paper, the sets D1 and D2 are closed intervals ␮ ␮ that generally ˜␭ ͑f͒Ϸ␭ ͑f͒, but we expect that the values and 1 and 2 are Lebesgue measures; for simplicity, we call O O ˜␭ ͑ ͒ ␭ ͑ ͒ ͑ this the standard case. Unless otherwise stated, we assume O f and O f will both change in the same direction ei- ␮ ␮ ͒ that 1 and 2 are finite and nonatomic measures. Addition- ther increase or decrease in response to a change in the ʚ n ʚ n ally, we assume that D1 R and D2 R , with topologies argument function f. In other words, the hallmarks of an

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͑ ͒ ␭ ͑ ͒ ͑ ͒ FIG. 8. ͑Color͒ Continuity defect. See definition of ␭ ,Eq.͑56͒. ͑Left͒ For FIG. 7. Color 1-1 defect. See definition of 1-1,Eq. 49 . Left A typical C ␮ a typical discontinuity, we ␮ measure the size of the jump ͑and then choose deficiency, where we take a 1 measure of the set that must be removed 2 ͑ ͒ the largest such jump͒. ͑Right͒ When D is not an interval, we must be more from D1 to make the remaining function 1-1 the blue horizontal segment 2 ␮ ͑ careful. The discontinuity in region A does not create a defect, because it lies and a 2 measure of the portion of D2 which is multiply covered the yellow͒. ͑Right͒ In this example, note ͑1͒ the nonmonotonicity near region A outside of D2, while in region B, we only measure the portion of I that lies is not measured, because it lies outside the sets of interest; ͑2͒ the horizontal inside D2. component in region B results in a defect from the horizontal measurement, but no contribution from the vertical; ͑3͒ in region C, we take the interval to the left of the jump to perform the horizontal measurement because of the D. Measuring discontinuities of f ␭ inf in the definition of 1-1. To measure discontinuities, we note that a continuous function maps “small sets to small sets,” and that a discon- appropriate suitable surrogate are such monotonicity with tinuity is indicated when an arbitrarily small set is mapped to a large one. In particular, we seek to identify when the f respect to parameter variations between ˜␭ and ␭. This may undergoes a “jump,” and measure the jump. To allow the not always hold true, but it is often true in application on the possibility of Cantor sets for D and D , we will need to simple examples we describe here. 1 2 define these ideas in terms of sets and measures of those sets. ෈ ␦Ͼ For each x0 D1 and for each 0, we define the set C. 1-1 deficiency B͑␦ ͒ ͕ ෈ ͉ ͉ Ͻ ␦͖ ͑ ͒ ,x0 ª x:x D1, x − x0 , 53 f To measure the extent to which is not 1-1, we need to which creates a nested family of sets as ␦0. We measure where quantify the function is not 1-1, by measurement on the f image of these sets by defining the domain of f, and the extent of the folding,27 a measurement on the range. We proceed as follows: we define G to be ␮ ͑I പ D ͒ ͑ ͒ 2 2 ͑ ͒ a␦ x0 inf . 54 the collection of all subsets GʚD which satisfy the follow- ª ʛ ͓B͑␦ ͔͒ ␮ ͑ ͒ 1 I f ,x0 2 D2 ing: Because a␦͑x ͒ is monotonically decreasing with decreasing ␮ 0 • G is 1 measurable. ␦, we can take the limit as ␦0, defining ͓ ͔ ␮ • f G is 2 measurable. ͑ ͒ ͑ ͒ ͑ ͒ • f is restricted to G is 1-1. a x0 ª lim a␦ x0 , 55 ␦→0+ ¯ 28 For any such G, we denote its complement in D1 by G where we think of a͑x ͒ as being the atomic part of f. We ϵ 0 D1 −G. Then we define the 1-1 defect by define ␭ ͑ ͒ ͑ ͒ ͑ ͒ ¯ ¯ C f sup a x0 . 56 ␮ ͑G͒ ␮ ͑f͓G͔͒ ª ෈ ␭ ͑ ͒ ͫ 1 2 ͬ ͑ ͒ x0 D1 1-1 f ª inf + . 49 GʚG 2␮ ͑D ͒ 2␮ ͑D ͒ 1 1 2 2 See Fig. 8. ␭ ͑ ͒ See Fig. 7. Because C f is fundamentally based on intervals, we In the standard case, we simply try to identify the largest generally find that it is sufficiently easy to approximate such part of the range that is multiply covered. We define enve- that we have not used a surrogate. However, we note that if lope functions D1 is an interval, we can define ˜␭ ͑ ͒ ʈ ͑ ͒ʈ ͑ ͒ e+͑x͒ = sup f͑y͒, ͑50͒ C f = a x0 p. 57 D ෉yഛx 1 ˜␭ ͑ ͒ϵ␭ ͑ ͒ ϱ We note that C f C f when p= , but the flexibility to use other norms might prove useful in some situations. e−͑x͒ = inf f͑y͒. ͑51͒ ෉ ജ D1 y x E. Discontinuity in f−1 Then e+͑x͒ records the largest function value to the left of x, while e−͑x͒ records the smallest function value to the right of In the standard case, we note that if f is 1-1 and con- −1 x. Then tinuous, then f is well defined and is also continuous, so this requirement on a conjugacy may seem redundant. Gen- erally, this requirement for a topological conjugacy is needed ˜␭ ͑f͒ = ʈe+͑x͒ − e−͑x͒ʈ . ͑52͒ 1-1 p because the domain space and range space may be defined We often choose p=ϱ, yielding the sup norm, but other with very different topologies. In our situation, although the choices for p may be useful in some situations. topologies are typically similar ͑based on inheriting the usual

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FIG. 9. ͑Color͒ These two tent maps are almost conjugate. ˜␭͑f͒ FIG. 10. ͑Color͒ Maps that are not close to conjugacy. ˜␭͑f͒Ϸ1/4͑0+0+0 Ϸ / ͑ ͒ ͑ ͒ 1 4 0.032+0+0.032+0 . Note that g1 1 0. +0.163͒. Near x=1/2, the commuter, f is actually horizontal, indicating that there is an interval of orbits of g1 than cannot be represented in g2, but are instead associated with the vertical sections of the green graphs of g2, which we defined as gˆ −1, which “complete” the inverse of g , as required by Eq. ͒ 2i 2 topology of the line , we find that directly measuring this ͑30͒. The apparent vertical gap of the commuter is not a defect of continuity, fourth defect is the easiest and most direct way to measure but is simply the result of an extremely steep section in the graph, and not gaps in the domain of definition. To define this defect, we enough points are plotted to fill in the picture. We know there is no actual employ the same strategy as for measuring the continuity of vertical gap because g1 is a full shift and is able to match all orbits of g2. f. For each y ෈D and for each ⑀Ͼ0, we define the set 0 2 commutation operator. For each commuter, the caption B͑⑀ ͒ ͕ ෈ ͉ ͉ Ͻ ⑀͖ ͑ ͒ ˜␭͑ ͒ / ͑˜␭ ˜␭ ,y0 ª y:y D2, y − y0 . 58 shows the approximate computation f =1 4 O + 1-1 +˜␭ +˜␭ −1͒. We measure the pre-image of these sets by defining C C

␮ ͑I പ D ͒ ͑ ͒ 1 1 ͑ ͒ VII. CONCLUSIONS aˆ ⑀ y0 ª inf . 59 ʛ −1͓B͑⑀ ͔͒ ␮ ͑D ͒ I f ,y0 1 1 In this paper, we have put forward a generalizable Taking the limit ⑀0, we define method based on functional fixed-point iteration to explicitly construct the change of coordinates function that acts as a ͑ ͒ ͑ ͒ ͑ ͒ conjugacy between two topologically conjugate dynamical aˆ y0 ª lim aˆ ⑀ y0 , 60 ⑀→0 systems. We expect that our method of fixed-point iteration ͑ ͒ −1 will extend, in a straightforward manner, to building conju- where we think of aˆ y0 as being the atomic part of f .We define gacy functions between higher dimensional systems when the symbol generating partitions are available. Of course, we ␭ ͑ ͒ ͑ ͒ ͑ ͒ 29,30 C−1 f sup aˆ y0 . 61 know that this last caveat is nontrivial, and this research ª ෈ y0 D2 is one aspect of our continuing work in the subject. In the standard case, we use this defect to measure gaps in While construction of conjugacy functions is interesting ␭ in its own right, we do not consider this to be the main the domain. Similarly as for C, a suitable surrogate is to measure the largest such gap in the domain. We define I to intellectual contribution of the work. In our opinion, a re- ʚ ͑ ͒ markable aspect of our methods of fixed-point iteration is be the set of all intervals I D1 such that f x is undefined or constant for all x෈I. Then we measure the defect as that it still produces a commuter, even between nonequivalent systems, and in a sensible manner according to choices ˜␭ ͑ ͒ ͑ ͒ ͑ ͒ C−1 f = sup m I , 62 IʚI where m is a Lebesgue measure. ␭ ͑ ͒ The picture for C−1 f is not shown, but would be simi- ␭ ͑ ͒ lar to that shown for C f in Fig. 8, where the roles of domain and range are reversed in the obvious manner for inverse functions.

VI. EXAMPLES

In this section, we present several examples of compar- FIG. 11. ͑Color͒ Logistic map with “just a little” extra dynamics. Although these maps are nearly the same height, they are far from conjugacy. That g1 ing maps g1 and g2 by showing the resulting commuter f, ⌺ ͑ ͒ has two humps, and g2 has one hump, which suggests that subshifts of 4 whether it be a homeomorphism or not Figs. 9–16 . Each ⌺ ˜␭͑ ͒ and 2, respectively, are required to represent the dynamics. f example follows the same presentation template: in the left Ϸ / ͑ ͒ 1 4 0.12+0.056+0.12+0 . However, g1 behaves as if it is almost conju- panel, we graph g1 in blue and g2 in red. Additionally, the gate to a trapezoid map ͑Ref. 37͒, also known as a gap-map ͑Ref. 38͒, since graphs for gˆ −1 are displayed in green circles. Along the hori- it does not take a large perturbation to replace the extra humps of x 2i ෈͑ ͒ zontal axis, the blue and red rectangles indicate the chosen 0.25,0.5 , with a horizontal line segment. However, our ﬁrst choice, using a tent map of approximately the same height as g , does not match intervals I and I . The right panel graphs the resultant 1 Xi Yi very well. Note the vertical gaps in the commuter, indicating that g2 has commuter function created by repeated application of the dynamics that are not matched by g1.

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͑ ͒ FIG. 13. Color Identical on the invariant set. For this example, g1 and g2 ͓ ͔ are identical on the interval 0.1, 1 . However, whereas g2 is a simple tent ͓ ͔ map, g1 has an additional hump on the interval 0, 0.1 . Because the maps are the same over a portion of the domain, the commuter coincides with the identify map on an interval. ˜␭͑f͒Ϸ1/4͑0+0.12+0+0͒. If dynamics were restricted to the invariant set of each system, then the two maps would be conjugate, with f͑x͒=x the homeomorphism.

pared by choice of a particular functional Banach space, such as L2, it is obvious and well known that similarity or dissimilarity within such a norm does not directly address comparison between orbits of the two systems. Instead, we have claimed that comparison of nonequivalent dynamical systems could be made via the commuter function by measuring how it fails to be a homeomorphism. We believe that the specific details of our defect cost function ␭ are reasonable and useful to our stated purposes in the manifesto of Sec. IV A. Furthermore, we have designed flexibility for the modeler to design the details of ␭ so as to focus on various aspects of the mismatch. As we have shown in the examples ͑ ͒ FIG. 12. Color Better projective comparison. The blue map g1 is identical of Sec. VI, a nonzero defect is generally descriptive of non- to that shown in Fig. 11, but here it is compared to a shorter tent map, matching between some of the orbits of the two dynamical ͑ ͒ g2 x =T0.55. Therefore, this g2 involves much less folding of the interval into systems, and the concept of measuring mismatch is the main itself than the g of Fig. 11. Because of the L1 difference between the maps, 2 issue we wish to emphasize, separate from the details of our we might presume g2 is less close to g1 in an almost conjugacy sense. ˜␭͑ ͒Ϸ / ͑⑀ defect functions. Thus, beyond entropy, we have outlined a However, our almost conjugacy surrogates measure f 1 4 1 +0.004 ⑀ ͒ ⑀ + 3 +0.031 , where i denotes a numerically small value, which is signifi- new method measuring the dissimilarity of the languages cantly smaller than the defect in Fig. 11. Although the graph seems to corresponding to orbit structures of two nonequivalent dy- ˜␭ indicate a large onto deficiency, the small O can be understood by first namical systems. In this sense, our new methods may be ͓ ͔ considering only the invariant sets of each map, D1 = 0.201,0.95 and D2 =͓0.495,0.55͔, instead of the unit intervals ͓0, 1͔ shown. This scenario is considered more natural, within the context of a dynamical ˜␭ systems-based comparison, than a direct comparison of the emphasized by the inset picture of f, where we can see that O is small, and the other defects are even smaller. Comparing the maps g1 and g2 outside of right-hand sides of dynamical systems by using the norm of the invariant sets, say on the left sides, near x=0, the maps are very similar: the difference in some Banach space. each sweeps initial conditions monotonically into D1 and D2, respectively. There are a number of theoretical and applications ori- Thus the extension of f to the full ͓0, 1͔ to ͓0, 1͔ does not suffer the apparent ˜␭ ented directions that we are currently pursuing in the future onto deficiency near x=0 and x=1. 1-1 is small because this surrogate measures only the difference between the upper and lower envelope, as per development of this work. These efforts include a topological ͑ ͒ ˜␭ Eq. 52 . f is not continuous, but C is small since the vertical steps are based parameter estimation scheme for modeling dynamical ˜␭ short. The largest defect measurement, C−1, comes from short horizontal systems within an assumed model classes, for example to components, which are measured as discontinuities in f−1. Why would the match “toy models” to observed data. Similarly, our methods much shorter tent map g2 here measure better as almost conjugate than the should be considered as a principled way to validate models g shown in Fig. 11? The result is reasonable when we focus on invariant 2 produced by analysis of time-delay embedded data, where sets: The blue curve on its invariant set is much more like the green g2 in this example than the g2 in Fig. 11, where a large fraction of the unit interval the primary desire is proper representation of the dynamical is folded over itself. We conclude that although the blue curve is tall, on its characteristics of the original system. Furthermore, since cer- invariant set, it appears to be much like a scaled, almost conjugate, version tain systems of differential equations ͑such as the Lorenz of the short tent map on its invariant set. equations͒ admit Poincaré surface of sections which are very much like one-dimensional maps, our techniques are already directly applicable to such ODEs. We are also pursuing an and preconceptions of the modeler. We put forward that the extension of our fixed-point iteration scheme to allow for major contribution of this work is the thesis that dynamical multivariate systems, within our ability to decide generating systems should be comparable in a sense that is consistent partitions. In this context, there is promise to extend our with the topological conjugacy notion of equivalence; this is, methods for comparison of general classes of differential after all, the centerpiece of the field of dynamical systems. equations. There is no fundamental roadblock to a multivari- Whereas nonequivalent systems have typically been com- ate extension of the fixed-point iteration scheme, and since

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͑ ͒ ͑ ͒ ͑ ͒ FIG. 14. Color The period–3 window. Choosing g1 to be the logistic map FIG. 16. Color The “too-tall” logistic map. The literature Refs. 5 and 9 with parameter 3.84, there is an attracting period–3 orbit whose basin is a tells us that the invariant set of this logistic map is semiconjugate to the full point in the interval. A Cantor set of initial conditions that are not in that tent map. This relationship is a conjugacy if the domain of the tent map basin constitute a chaotic saddle of g1. We compare this map to a submaxi- excludes the peak point, and all of its pre-images. This commuter f, shown mal tent map, where we have chosen the height of that tent map so that the here for the ﬁrst time, has defects ˜␭͑f͒Ϸ1/4͑0+0+0+1/3͒, and it is, in defect will be small, with ˜␭͑f͒Ϸ1/4͑0.0005+0+0.0005+0.022͒. A higher fact, a devil staircase function with ﬂat spots over the holes of a Cantor set. tent map would create large intervals on which f is horizontal, while choos- Notice the strong similarity to the previous two examples. ing a smaller tent would create larger vertical gaps. Each horizontal component indicates that f is associating an interval of g1 dynamics to a single point in g2. understand the relationship between dynamical systems g1 and g2 within the context of the commutative diagram: the design of the defect functions was entirely measure- based, it will extend directly to this setting. A future goal for this work is to be able to measure the degree to which a toy model is descriptive of the larger system ͑such as a Galerkin projection of a PDE͒.39,40

ACKNOWLEDGMENTS This research was supported under NSF Grant No. DMS-0404778. We thank the anonymous referees for their g thorough feedback and thoughtful suggestions. 1 X ——→ X f ↓↓f ͑A1͒ APPENDIX A: QUADWEBBING g2 Y ——→ Y “Quadwebbing” is our name for a graphical representation that allows us to visualize the action of the both com- Each quadweb is constructed by dividing the figure into a mutation operator and the commuter in relation to one- two-by-two panel of axes, where each panel shows the graph dimensional ͑1D͒ maps of the interval. The basic structure of of one of the four functions shown in the commutative dia- a quadweb diagram is based on the idea that we seek to gram. The plots are arranged to allow a graphical illustration of how the function f satisfies the commutative diagram. Figure 1 ͑left͒ illustrates the case of conjugate dynamical systems. In addition to illustrating how the commuter satisfies Eq. ͑A1͒, it reveals where it might fail. Specifically, as we recall from Sec. III, we noted that to have a well defined operator, we needed to extend the graph of g2 such that its inverse could be applied to any y෈Y. We denoted the modified −1 31 graphs as gˆ 2i . We use the term The zed describes that portion of the graph of ͑ ͒ −1 FIG. 15. Color Two full-shift maps that are not conjugate. g1 is a slight gˆ that does not coincide with g . ͑A2͒ alteration of the full-shift symmetric tent. The right leg has been modified so 2i 2 ͒ ͑ ͒ ͑ that a small portion has a slope m, with ͉m͉Ͻ1, where that interval contains ؠ If f g1 x lies on the zed, then Eq. A1 does not hold for the fixed point. Therefore, g1 has a stable fixed point at xf =0.665. This commuter is similar in character to that of Fig. 14—the stable periodic point that x. This phenomenon can be interpreted to mean that g1 of the X dynamics means that an entire interval of initial conditions must be has more dynamics than can be represented by g2 under the associated with a single point in the Y dynamics. Since the initial condition assumed partitions. Figure 17 illustrate this behavior. is in the basin of attraction for this fixed point, the resultant commuter is a The quadweb diagram allows for easy understanding of ˜␭͑ ͒Ϸ / ͑ ͒ devil’s staircase function. f 1 4 0+0+0+0.048 . We remark that when what some typical defects imply about the dynamics of the measuring the defect via suitable surrogate, the deficiency caused by horizontal portions of the commuter is recorded as a defect in the continuity of system. In particular, wherever f is horizontal, we see an the inverse, not by the defect in 1-1. interval of initial conditions for system g1 that must be rep-

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͑ ͒ ͑ ͒ FIG. 17. Color Quadweb for nonconjugate maps. g1 maps across X a full FIG. 18. Color Quadweb illustration of the commutation operator. For three times. g2, shown in red, also has three laps, but they do not map ease of illustration, we choose f0 as the identify map, and plot it in the lower −1 completely across Y. The yellow graph shows the gˆ 2i , which are used to left corner. The orange vertical shows a chosen x coordinate, and the green ͒ ͑ ؠ ؠ −1 construct a well-defined commutation operator. The green path illustrates path shows the graphical computation of g2 f g1 x . The intersection in the ؠ g2 f g1 for a particular x. Using the same x, we use the purple path to show upper right quad is a point that lies on f1 =Cg f0. Generally, you would need ؠ ͑ ͒ 1 g2 f x , which obviously does not coincide with the green. Rather the green to repeat for a large number of x coordinates. However, since all maps are path shows that we land on the zed which is the case for all x inside the linear in this example, a few well chosen points are sufficient to describe the intervals colored pink. The black rectangles illustrate that Eq. ͑A1͒ holds at first application of the commutation operator. x lying outside these intervals.

highlighted in this paper, and some insight into what may resented by a single point under g . Similarly ͓see Fig 1 2 become of the geometry of nonhomeomorphic commuters. ͑right͔͒ when f has a vertical gap, there is an interval in g 2 We show that the conjugacy function h͑x͒ which results is that has no associated orbit in system g . 1 both a Lebesgue singular function, and in many ways re- As a final illustration of alternative uses of the quadweb, minds us of a de Rham function. The similarity of our ho- we use the diagram to provide a graphical description of the meomorphisms h͑x͒ to the de Rham functions arises from the action of the commutation operator. If we graph an arbitrary fact that both are solutions to functional equations of similar f0 in the lower left panel, we can graphically compute f1 g2 form. We note that required solutions to de Rham–like equa- =C f0 as follows: Choose an arbitrary x coordinate in the g1 tions can arise quite naturally in other settings of dynamical right half of the quadweb. Move up until you reach g , then 1 systems,34 with a standard solution approach of finding an left until you reach f , up again to g , then to the right. 0 2 expression of the problem that can be solved using the Con- Although we used g and f in the usual way of graphically 1 0 traction Mapping Principle. applying a function, we note that we used graph g by going 2 We recall definitions of these peculiar functions before from range Y to domain Y, equivalent to applying g=1. Con- 2 we proceed to prove the properties of our h͑x͒. While we .sequently, we graphically “compute” the path g−1ؠ f ؠg ͑x͒ 2 1 have not proven that homeomorphisms between two tents The resultant intersection of that line with the original x co- that are not both full have these properties, and we also do ordinate is a point on f . Repeating at a sufficiently dense set 1 not prove these properties for either nontent maps or nonfull of x coordinates will allow a robust description of f . See 1 shift tent maps, the properties are certainly suggestive of the Fig. 18 for an example of this technique. nature of the kinds of peculiarities one might expect, and even which seem apparent in the simulations. Most striking APPENDIX B: RELATIONSHIP TO DE RHAM FUNCTIONS AND LEBESGUE SINGULAR FUNCTIONS is the degree to which we now know that the usual homeomorphism example between a full tent and the full logistic In this appendix, we will give a direct and constructive map, which gives rise to a diffeomorphism, can mislead in- proof of the singular nature of homeomorphisms between tuition from the more typical conjugacy. full shift tent maps. A standard result from ergodic theory gives that probability measures on distinct shift maps are 1. Lebesgue singular functions mutually singular.32 Since the conjugacy must map the uniform measure of the symmetric tent to the mutually singular Definition [36]: A continuous function of bounded varia- probability measure on a skew tent, that conjugacy must be a tion s͑x͒ is called a singular function if it is differentiable singular function.33 The purpose of including a proof of the almost everywhere on its domain, and the derivative sЈ͑x͒ special case form of this fact, in the form of Proposition 1 =0 where it exists, and it is called a Lebesgue singular func- below, is that we hope that its constructive approach gives tion when almost every one is in the sense of a Lebesgue explicit intuition for general homeomorphisms, to the geo- measure. metric constructive nature of the fixed-point iteration scheme Definition [39 and 40]: de Rham functions. A de Rham

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function is the unique bounded solution of the functional k k +1 x ෈ I ͩ n , n ͬ; ͑B5͒ equation n ª 2n 2n t t +1 measuring the range of h over the interval I , we define ␾ͩ ͪ = a␾͑t͒, ␾ͩ ͪ = a + ͑1−a͒␾͑t͒, ͑B1͒ n 2 2 k +1 k ͩ n ͪ ͩ n ͪ ͑ ͒ pn h − h . B6 where a෈͑0,1͒ is fixed and Eq. ͑B1͒ is satisfied for all ª 2n 2n t෈͓0,1͔. Since hЈ͑x͒ exists, we know that p 2. h x is a singular function h͓͑k ͒2−n,͑k +1͒2−n͔ = n → hЈ͑x͒, ͑B7͒ „ … n n 2−n In this section, we will show that for some parameter where the left-hand side symbolizes the Newton divided dif- values, the f␯ that results from Lemma 3 is a singular func- ference. If hЈ͑x͒0, then the ratio of two successive divided tion. The argument of this proof follows closely the structure differences ͑for n and n+1͒ will tend to 1. Therefore, the found in Ref. 19, which is standard for the de Rham func- ratio tions. However, slight modifications to the proof are required due to the differences between his problem and ours. We will p 1 n+1 → ͑ ͒ specifically consider the conjugacy between a full shift skew , B8 pn 2 tent map and the full shift symmetric tent. as long as hЈ͑x͒ is distinct from 0. Consider the set of skew tent maps of the form Sa,1, which are the maps that generate a full shift on two symbols. To complete the proof, we will show that either / / These maps are conjugate to the full shift symmetric map, pn+1 pn =a or pn+1 pn =1−a, where this property holds for T , by which we infer that ␯=␯ =1. If a=1/2, then SϵT , all x. This portion of proof results from the self-similar struc- 1 0 1 ͑ ͒ and the conjugacy would be the identify map. However, ture that derives from the fact that h x is the fixed point of ͑ ͒ when a1/2, we can find the conjugacy h by developing the contraction mapping M1 defined by Eq. 13 , where we ␯ ␯ the contraction operator and identifying its fixed point. Then are using b= = 0 =1. The proof proceeds by induction: f␯ ϵ f␯ ϵ f ϵh. Because h is a conjugacy, we know it is ͑ ͒ ͑ ͒ 0 1 • True for n=0. We have p0 =h 1 −h 0 =1−0=1. If x strictly increasing and continuous. However, we will find ഛ / ͑ / ͔ ͑ / ͒ / 1 2, then I1 = 0,1 2 ;h 1 2 =a, and p1 p0 =a. For x that despite these restrictions, h remains a very strange func- Ͼ / ͑ ͒ ͑ / ͒ / 1 2,p1 =h 1 −h 1 2 , giving p1 p0 =1−a. tion. • Induction. Assume that for arbitrary x, p /p 35,36 j+1 j Recall that by the Lebesgue Decomposition Theo- ෈͕a,1−a͖ for all 0ഛ jഛn−1. We will show that the prop- rem, a function g͑x͒ of bounded variation may be decom- erty holds for j=n. posed, / n Note that the end points of In are xleftªkn 2 and xright ͑ ͒ϵ ͑ ͒ ͑ ͒ ͑ ͒ / n g x a x + s x , B2 ªkn +1 2 . Denote the midpoint of In as xmid. Then either I =͑x ,x ͔͑which implies that k =2k ͒ or I where a͑x͒ is absolutely continuous and s͑x͒ is purely singu- n+1 left mid n+1 n n+1 =͑x ,x ͒, implying k =2k +2. If we assume that I lar. As an additional description of this decomposition, we mid right n+1 n n+1 is on the left half of I , then have n ͩ 2kn +1ͪ ͩ 2kn ͪ h n+1 − h n+1 ͑ ͒ Ј͑ ͒ ⇒ ͑ ͒ ͑ ͒ Ј͑ ͒ ͑ ͒ pn+1 2 2 a x = ͵ g x dx g x = s x + ͵ g x dx. B3 = , ͑B9͒ p k +1 k n hͩ n ͪ − hͩ n ͪ So the singular part separates out the components that pre- 2n 2n vent a function from satisfying the fundamental theorem of whereas if I is the right half of I , we have calculus. Because h is strictly monotone and continuous on a n+1 n closed interval, it is of bounded variation, so it is decompos- 2k +2 2k +1 hͩ n ͪ hͩ n ͪ able as above. However, in the following paragraphs, we n+1 − n+1 pn+1 2 2 show that h͑x͒ is itself a purely singular function, such that = . ͑B10͒ pn ͩ kn +1ͪ ͩ kn ͪ h n − h n h͑x͒ϵa͑x͒ + s͑x͒ ⇒ a͑x͒ϵ0. ͑B4͒ 2 2 ͑ ͒ We now take advantage of the self-similarity implied by Eq. Proposition 2. Let h x be the conjugacy between Sa,1 ෈S / ͑ ͒ ͑7͒: When xഛ1/2, we have that ah͑2x͒=h͑x͒. So for each and T1, where a 1 2. Then hЈ x exists for almost every x෈͓0,1͔. Moreover, hЈ͑x͒=0 wherever hЈ͑x͒ exists. function evaluation in Eqs. ͑B9͒ and ͑B10͒, we perform the Proof. Because h must be a homeomorphism, it is in- substitution 36 creasing. Then a standard result from analysis tells us that m m h is a.e. differentiable, giving the existence of hЈ͑x͒. Suppose ahͩ ͪ = hͩ ͪ. ͑B11͒ 2r−1 2r that the derivative exists at some 0ϽxϽ1; we will show that Ј͑ ͒ Ј h x =0. To simplify notation, we denote by In the dyadic intervals Ј For each n, we find integer kn such that containing the point 2x, and define pn as the length of the

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jsp 013118-17 Homeomorphic defect of modeling Chaos 18, 013118 ͑2008͒ intervals h͓IЈ͔. Then applying Eq. ͑B11͒ to Eqs. ͑B9͒ and 8E. Ottt, Chaos in Dynamical Systems, 2nd ed. ͑Cambridge University n ͒ ͑B10͒, we have that Press, Cambridge, UK, 2002 . 9C. Robinson, Dynamical Systems; Stability, Symbolic Dynamics, and ͑ ͒ 2k +1 2k Chaos, 2nd ed. CRC Press, Boca Raton, FL, 1999 . ahͩ n ͪ − ahͩ n ͪ 10E. N. Lorenz, “A study of the predictability of a 28-variable atmospheric n n Ј ͑ ͒ pn+1 2 2 pn model,” Tellus 17 321 1965 . = = , ͑B12͒ 11 p k +1 k pЈ E. N. Lorenz and K. A. Emanuel, Optimal Sites for Supplementary n ͩ n ͪ ͩ n ͪ n−1 Weather Observations: Simulation with a Small Model ͑American Meteo- ah n−1 − ah n−1 2 2 rological Society, Washington, DC, 1998͒, p. 399. 12D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Cod- or ing ͑Cambridge University Press, New York, 1995͒. 13K. Thompsen, “The defect of factor maps and finite equivalence of dy- 2k +2 2k +1 namical systems,” Topics in Dynamics and Ergodic Theory, edited by S. ahͩ n ͪ − ahͩ n ͪ p 2n 2n pЈ Bezuglyi and S. Kolyada ͑Cambridge University Press, Cambridge, UK, n+1 = = n . ͑B13͒ 2003͒, pp 190–225. p k +1 k pЈ 14L. Bunivmovich et al., Dynamical Systems, Ergodic Theory, and Applica- n ahͩ n ͪ − ahͩ n ͪ n−1 2n−1 2n−1 tions, 2nd ed., edited by Y. G. Sinai ͑Springer-Verlag, Berlin, 2000͒. 15G.-C. Yuan, J. A. Yorke, T. L. Carroll, E. Ott, and L. M. Pecora, “Testing By assumption, whether two chaotic one dimensional processes are dynamically identical,” Phys. Rev. Lett. 8520, 4265 ͑2000͒. 16 pЈ M. Shub, “Endomorphism of compact differentiable manifolds,” Am. J. n ෈ ͕a,1 − a͖. ͑B14͒ Math. 91, 175 ͑1969͒. Ј 17 pn−1 M. Shub and D. Sullivan, “Expanding endomorphisms of the circle revis- ited,” Ergod. Theory Dyn. Syst. 5, 285 ͑1985͒. The argument for xϾ1/2 is similar, though the scaling factor 18W. de Melo and S. van Strien, One-Dimensional Dynamics ͑Springer- is 1−a instead of a. ᮀ Verlag, Berlin, 1992͒. 19P. Billingsley, Probability and Measure, 3rd ed. ͑Wiley-Interscience, New York, 1995͒. 3. Remarks and lessons from the de Rham–like 20M. Misiurewicz and E. Visinescu, “Kneading sequences of skew tent conjugacies maps,” Ann. I.H.P. Probab. Stat. 27, 125 ͑1991͒. 21J. P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attrac- We have remarked that since two piecewise linear maps tors,” Rev. Mod. Phys. 57, 617 ͑1985͒. with different metric entropies must not be diffeomorphically 22J. Hale, Ordinary Differential Equations ͑Kreiger, Melbourne, FL, 1980͒. 23 related, then any conjugacy between two such maps cannot The major step of choosing and matching partitions of the two dynamical systems is somewhat easier in one-dimensional phase spaces, with inter- be everywhere differentiable. It may not be immediately ob- esting and important consequences if for some reason the generating par- vious what is the distinguished point or points where the titions are not matched, as studied in our previous work ͑Ref. 29͒.Itmay discontinuity of the derivative of h͑x͒ should reside. Now, in often be useful for complexity reduction of the models to deliberately not light of the analysis of the de Rham like properties of h͑x͒ in choose generating partitions, as exemplified by the examples in Figs. 11 the previous section, we see that the discontinuities are and 12. In multivariate dynamical system, choosing the partitions to com- Ј pare will be a challenging part of the problem in future work. For ex- dense, located at endpoints of the dyadic intervals In, corre- ample, the generating partition for certain diffeomorphisms of the plane sponding to preimages of the map peaks, where there is a are conjectured to coincide to a curve connecting “primary” heteroclinic discontinuity in the slope of the nth composition of the sym- tangencies ͑Ref. 30͒ but again there is useful information to be gained by matching without regard to the generating partition ͑Ref. 29͒. metric tent map. 24 ෈ ͑ ͑ ͒͒෈ ϫ Cover—with each y Y we associate a point y,g2 y Y Y. The collection of all such points is called the graph of g . A point ͑y,g ͑y͒͒ on 1 2 2 D. K. Arrowsmith and C. M. 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