DOCSLIB.ORG

A Concept of Homeomorphic Defect for Defining Mostly Conjugate

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 func- tional 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 prin- cipled 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 conju- gacy), 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 1054-1500/2008/18͑1͒/013118/18/$23.0018, 013118-1 © 2008 American Institute of Physics Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jsp 013118-2 J. D. Skufca and E. M. Bollt Chaos 18, 013118 ͑2008͒ 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.
Load more

Recommended publications
  • The Topological Entropy Conjecture
    mathematics Article The Topological Entropy Conjecture Lvlin Luo 1,2,3,4 1 Arts and Sciences Teaching Department, Shanghai University of Medicine and Health Sciences, Shanghai 201318, China; [email protected] or [email protected] 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China 3 School of Mathematics, Jilin University, Changchun 130012, China 4 School of Mathematics and Statistics, Xidian University, Xi’an 710071, China Abstract: For a compact Hausdorff space X, let J be the ordered set associated with the set of all finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with ¶. Therefore, we have Hˇ p(X; Z), where 0 ≤ p ≤ n = nJ. For a continuous self-map f on X, let a 2 J be f an open cover of X and L f (a) = fL f (U)jU 2 ag. Then, there exists an open fiber cover L˙ f (a) of X induced by L f (a). In this paper, we define a topological fiber entropy entL( f ) as the supremum of f ent( f , L˙ f (a)) through all finite open covers of X = fL f (U); U ⊂ Xg, where L f (U) is the f-fiber of − U, that is the set of images f n(U) and preimages f n(U) for n 2 N. Then, we prove the conjecture log r ≤ entL( f ) for f being a continuous self-map on a given compact Hausdorff space X, where r is the maximum absolute eigenvalue of f∗, which is the linear transformation associated with f on the n L Cechˇ homology group Hˇ ∗(X; Z) = Hˇ i(X; Z).
    [Show full text]
  • Topological Invariance of the Sign of the Lyapunov Exponents in One-Dimensional Maps
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 1, Pages 265–272 S 0002-9939(05)08040-8 Article electronically published on August 19, 2005 TOPOLOGICAL INVARIANCE OF THE SIGN OF THE LYAPUNOV EXPONENTS IN ONE-DIMENSIONAL MAPS HENK BRUIN AND STEFANO LUZZATTO (Communicated by Michael Handel) Abstract. We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy. 1. Statement of results In this paper we consider C3 interval maps f : I → I,whereI is a compact interval. We let C denote the set of critical points of f: c ∈C⇔Df(c)=0.We shall always suppose that C is finite and that each critical point is non-flat: for each ∈C ∈ ∞ 1 ≤ |f(x)−f(c)| ≤ c ,thereexist = (c) [2, )andK such that K |x−c| K for all x = c.LetM be the set of ergodic Borel f-invariant probability measures. For every µ ∈M, we define the Lyapunov exponent λ(µ)by λ(µ)= log |Df|dµ. Note that log |Df|dµ < +∞ is automatic since Df is bounded. However we can have log |Df|dµ = −∞ if c ∈Cis a fixed point and µ is the Dirac-δ measure on c. It follows from [15, 1] that this is essentially the only way in which log |Df| can be non-integrable: if µ(C)=0,then log |Df|dµ > −∞. The sign, more than the actual value, of the Lyapunov exponent can have signif- icant implications for the dynamics. A positive Lyapunov exponent, for example, indicates sensitivity to initial conditions and thus “chaotic” dynamics of some kind.
    [Show full text]
  • Writing the History of Dynamical Systems and Chaos
    Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as
    [Show full text]
  • Multiple Periodic Solutions and Fractal Attractors of Differential Equations with N-Valued Impulses
    mathematics Article Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses Jan Andres Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic; [email protected] Received: 10 September 2020; Accepted: 30 September 2020; Published: 3 October 2020 Abstract: Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5). Keywords: impulsive differential equations; n-valued maps; Hutchinson-Barnsley operators; multiple periodic solutions; topological fractals; Devaney’s chaos on attractors; Poincaré operators; Nielsen number MSC: 28A20; 34B37; 34C28; Secondary 34C25; 37C25; 58C06 1. Introduction The theory of impulsive differential equations and inclusions has been systematically developed (see e.g., the monographs [1–4], and the references therein), among other things, especially because of many practical applications (see e.g., References [1,4–11]).
    [Show full text]
  • Arxiv:1812.04689V1 [Math.DS] 11 Dec 2018 Fasltigipisteeitneo W Oitos Into Foliations, Two Years
    FOLIATIONS AND CONJUGACY, II: THE MENDES CONJECTURE FOR TIME-ONE MAPS OF FLOWS JORGE GROISMAN AND ZBIGNIEW NITECKI Abstract. A diffeomorphism f: R2 →R2 in the plane is Anosov if it has a hyperbolic splitting at every point of the plane. The two known topo- logical conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations (the existence of Anosov structures for plane translations was originally shown by W. White). P. Mendes con- jectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. We prove that this claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant under a time one map. 1. Introduction A diffeomorphism f: M →M of a compact manifold M is called Anosov if it has a global hyperbolic splitting of the tangent bundle. Such diffeomor- phisms have been studied extensively in the past fifty years. The existence of a splitting implies the existence of two foliations, into stable (resp. unsta- ble) manifolds, preserved by the diffeomorphism, such that the map shrinks distances along the stable leaves, while its inverse does so for the unstable ones. Anosov diffeomorphisms of compact manifolds have strong recurrence properties. The existence of an Anosov structure when M is compact is independent of the Riemann metric used to define it, and the foliations are invariants of topological conjugacy. By contrast, an Anosov structure on a non-compact manifold is highly dependent on the Riemann metric, and the recurrence arXiv:1812.04689v1 [math.DS] 11 Dec 2018 properties observed in the compact case do not hold in general.
    [Show full text]
  • Quasiconformal Homeomorphisms in Dynamics, Topology, and Geometry
    Proceedings of the International Congress of Mathematicians Berkeley, California, USA, 1986 Quasiconformal Homeomorphisms in Dynamics, Topology, and Geometry DENNIS SULLIVAN Dedicated to R. H. Bing This paper has four parts. Each part involves quasiconformal homeomor­ phisms. These can be defined between arbitrary metric spaces: <p: X —* Y is Ä'-quasiconformal (or K qc) if zjf \ _ r SUP 1^0*0 ~ Piv) wnere \x - y\ = r and x is fixed r_>o inf \<p(x) — <p(y)\ where \x — y\ = r and x is fixed is at most K where | | means distance. Between open sets in Euclidean space, H(x) < K implies <p has many interesting analytic properties. (See Gehring's lecture at this congress.) In the first part we discuss Feigenbaum's numerical discoveries in one-dimen­ sional iteration problems. Quasiconformal conjugacies can be used to define a useful coordinate independent distance between real analytic dynamical systems which is decreased by Feigenbaum's renormalization operator. In the second part we discuss de Rham, Atiyah-Singer, and Yang-Mills the­ ory in its foundational aspect on quasiconformal manifolds. The discussion (which is joint work with Simon Donaldson) connects with Donaldson's work and Freedman's work to complete a nice picture in the structure of manifolds— each topological manifold has an essentially unique quasiconformal structure in all dimensions—except four (Sullivan [21]). In dimension 4 both parts of the statement are false (Donaldson and Sullivan [3]). In the third part we discuss the C-analytic classification of expanding analytic transformations near fractal invariant sets. The infinite dimensional Teichmüller ^spare-ofnsuch=systemrìs^embedded=in^ for the transformation on the fractal.
    [Show full text]
  • Mostly Conjugate: Relating Dynamical Systems — Beyond Homeomorphism
    Mostly Conjugate: Relating Dynamical Systems — Beyond Homeomorphism Joseph D. Skufca1, ∗ and Erik M. Bollt1, † 1 Department of Mathematics, Clarkson University, Potsdam, New York (Dated: April 24, 2006) A centerpiece of Dynamical Systems is comparison by an equivalence relationship called topolog- ical conjugacy. Current state of the field is that, generally, there is no easy way to determine if two systems are conjugate or to explicitly find the conjugacy between systems that are known to be equivalent. We present a new and highly generalizable method to produce conjugacy functions based on a functional fixed point iteration scheme. In specific cases, we prove that although the conjugacy function is strictly increasing, it is a.e. differentiable, with derivative 0 everywhere that it exists — a Lebesgue singular function. When applied to non-conjugate 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 non-homeomorphic 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. We introduce methods to compare nonequivalent sys- tems by quantifying how much the commuter functions 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 includes fundamental issues as a principled understanding of the degree to which a “toy model” might be representative of a more complicated system, an important concept to clarify since it is often used loosely throughout science.
    [Show full text]
  • Fractal Geometry and Dynamics
    FRACTAL GEOMETRY AND DYNAMICS MOOR XU NOTES FROM A COURSE BY YAKOV PESIN Abstract. These notes were taken from a mini-course at the 2010 REU at the Pennsylvania State University. The course was taught for two weeks from July 19 to July 30; there were eight two-hour lectures and two problem sessions. The lectures were given by Yakov Pesin and the problem sessions were given by Vaughn Cli- menhaga. Contents 1. Introduction 1 2. Introduction to Fractal Geometry 2 3. Introduction to Dynamical Systems 5 4. More on Dynamical Systems 7 5. Hausdorff Dimension 12 6. Box Dimensions 15 7. Further Ideas in Dynamical Systems 21 8. Computing Dimension 25 9. Two Dimensional Dynamical Systems 28 10. Nonlinear Two Dimensional Systems 35 11. Homework 1 46 12. Homework 2 53 1. Introduction This course is about fractal geometry and dynamical systems. These areas intersect, and this is what we are interested in. We won't be able to go deep. It is new and rapidly developing.1 Fractal geometry itself goes back to Caratheodory, Hausdorff, and Besicovich. They can be called fathers of this area, working around the year 1900. For dynamical systems, the area goes back to Poincare, and many people have contributed to this area. There are so many 119 June 2010 1 2 FRACTAL GEOMETRY AND DYNAMICS people that it would take a whole lecture to list. The intersection of the two areas originated first with the work of Mandelbrot. He was the first one who advertised this to non-mathematicians with a book called Fractal Geometry of Nature.
    [Show full text]
  • Quasiconformal Homeomorphisms and Dynamics III: the Teichmüller Space of a Holomorphic Dynamical System
    Quasiconformal Homeomorphisms and Dynamics III: The Teichm¨uller space of a holomorphic dynamical system Curtis T. McMullen and Dennis P. Sullivan ∗ Harvard University, Cambridge MA 02138 CUNY Grad Center, New York, NY and SUNY, Stony Brook, NY 20 November, 1993 Contents 1 Introduction.................................... 2 2 Statementofresults ............................... 3 3 Classificationofstableregions. 8 4 The Teichm¨uller space of a holomorphic dynamical system . ......... 11 5 FoliatedRiemannsurfaces . .. .. .. .. .. .. .. .. 19 6 The Teichm¨uller space of a rational map . 22 7 Holomorphic motions and quasiconformal conjugacies . 28 8 Hyperbolicrationalmaps ............................ 36 9 Complex tori and invariant line fields on the Julia set . 37 10 Example: Quadraticpolynomials . 39 ∗ Research of both authors partially supported by the NSF. 1 1 Introduction This paper completes in a definitive way the picture of rational mappings begun in [30]. It also provides new proofs for and expands upon an earlier version [46] from the early 1980s. In the meantime the methods and conclusions have become widely used, but rarely stated in full generality. We hope the present treatment will provide a useful contribution to the foundations of the field. Let f : C C be a rational map on the Riemann sphere C, with degree d>1, whose → iterates are to be studied. ! ! ! The sphere is invariantly divided into the compact Julia set J, which is dynamically chaotic and usually fractal, and the Fatou set Ω= C J consisting of countably many open − connected stable regions each of which is eventually periodic under forward iteration (by the ! main theorem of [44]). The cycles of stable regions are of five types: parabolic, attractive and superattractive basins; Siegel disks; and Herman rings.
    [Show full text]
  • ERGODIC and SPECTRAL PROPERTIES an Interval Exchange
    STRUCTURE OF THREE-INTERVAL EXCHANGE TRANSFORMATIONS III: ERGODIC AND SPECTRAL PROPERTIES SEBASTIEN´ FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI ABSTRACT. In this paper we present a detailed study of the spectral/ergodic properties of three- interval exchange transformations. Our approach is mostly combinatorial, and relies on the dio- phantine results in Part I and the combinatorial description in Part II. We define a recursive method of generating three sequences of nested Rokhlin stacks which describe the system from a measure- theoretic point of view and which in turn gives an explicit characterization of the eigenvalues. We obtain necessary and sufficient conditions for weak mixing which, in addition to unifying all previ- ously known examples, allow us to exhibit new examples of weakly mixing three-interval exchanges. Finally we give affirmative answers to two questions posed by W.A. Veech on the existence of three- interval exchanges having irrational eigenvalues and discrete spectrum. 1. INTRODUCTION An interval exchange transformation is given by a probability vector (α1, α2, . , αk) together with a permutation π of {1, 2, . , k}. The unit interval [0, 1) is partitioned into k sub-intervals of lengths α1, α2, . , αk which are then rearranged according to the permutation π. Katok and Stepin [20] used interval exchanges to exhibit a class of systems having simple continuous spectrum. Interval exchange transformations have been extensively studied by several people including Keane [21] [22], Keynes and Newton [23], Veech [31] to [36], Rauzy [28], Masur [24], Del Junco [14], Boshernitzan [5, 6], Nogueira and Rudolph [27], Boshernitzan and Carroll [7], Berthe,´ Chekhova, and Ferenczi [4], Chaves and Nogueira [10] and Boshernitzan and Nogueira [8].
    [Show full text]
  • Abstract Similarity, Fractals and Chaos
    Abstract Similarity, Fractals and Chaos The art of doing mathematics consists in finding that special case which contains all the germs of generality. David Hilbert Marat Akhmet∗ and Ejaily Milad Alejaily Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey To prove presence of chaos for fractals, a new mathematical concept of abstract similarity is introduced. As an example, the space of symbolic strings on a finite number of symbols is proved to possess the property. Moreover, Sierpinski fractals, Koch curve as well as Cantor set satisfy the definition. A similarity map is introduced and the problem of chaos presence for the sets is solved by considering the dynamics of the map. This is true for Poincar´e,Li-Yorke and Devaney chaos, even in multi-dimensional cases. Original numerical simulations which illustrate the results are delivered. 1 Introduction Self-similarity is one of the most important concepts in modern science. From the geometrical point of view, it is defined as the property of objects whose parts, at all scales, are similar to the whole. Dealing with self-similarity goes back to the 17th Century when Gottfried Leibniz introduced the notions of recursive self-similarity [1]. Since then, history has not recorded any thing about self-similarity until the late 19th century when Karl Weierstrass introduced in 1872 a function that being everywhere continuous but nowhere differentiable. The graph of the Weierstrass function became an example of a self-similar curve. The set constructed by Georg Cantor in 1883 is considered as the most essential and influential self similar set since it is a simple and perfect example for theory and applications of this field.
    [Show full text]
  • Lecture 4. Entropy and Markov Chains
    preliminary version : Not for di®usion Lecture 4. Entropy and Markov Chains The most important numerical invariant related to the orbit growth in topological dynamical systems is topological entropy.1 It represents the exponential growth rate of the number of orbit segments which are distinguishable with an arbitrarily high but ¯nite precision. Of course, topological entropy is invariant by topological conjugacy. For measurable dynamical systems, an entropy can be de¯ned using the invariant measure. It gives an indication of the amount of randomness or complexity of the system. The relation between measure{theoretical entropy and topological entropy is given by a variational principle. 4.1 Topological Entropy We will follow the de¯nition given by Rufus Bowen in [Bo, Chapter 4]. Let X be a compact metric space. De¯nition 4.1 Let S ½ X, n 2 N and " > 0. S is a (n; "){spanning set if for every x 2 X there exists y 2 S such that d(f j(x); f j(y)) · " for all 0 · j · n. It is immediate to check that the compactness of X implies the existence of ¯nite spanning sets. Let r(n; ") be the least number of points in an (n; "){spanning set. If we bound the time of observation of our dynamical system by n and our precision in making measurements is " we will see at most r(n; ") orbits. Exercise 4.2 Show that if X admits a cover by m sets of diameter · " then r(n; ") · mn+1. De¯nition 4.3 The topological entropy htop(f) of f is given by 1 htop(f) = lim lim sup log r(n; ") : [4:1] "!0 n!+1 n In the previous de¯nition one cannot replace lim sup with lim since there exist examples of maps for which the limit does not exist.
    [Show full text]