DOCSLIB.ORG

Applications of the Birkhoff Ergodic Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Applications of the Birkhoff Ergodic Theorem APPLICATIONS OF THE BIRKHOFF ERGODIC THEOREM WEIAN WANG Abstract. Ergodic theory studies the long-term averaging properties of measure- preserving dynamical systems. In this paper, we state and present a proof of the ergodic theorem due to George Birkhoff, who observed the asymptotic equivalence of the time-average and space-average of a point x in a finite measure space. Then, we examine a number of applications of this theorem in number-theoretic problems, including a study of normal numbers and of L¨urothseries transformations. Contents 1. Recurrence and Ergodicity in Dynamical Systems 2 1.1. Approximation with Sufficient Semi-rings 3 2. The Birkhoff Ergodic Theorem 4 3. L¨urothSeries Transformations 8 3.1. Ergodic Properties of L¨urothtransformation T 10 4. The Normal Number Theorem 13 Acknowledgements 17 List of Figures 17 References 17 A discrete dynamical system consists of a space X and a transformation T which maps the space onto itself i.e. T : X X. We assume T is a measurable function. When studying discrete systems, we! consider how a point in the space moves over discrete time intervals. We can think of the X as the space of all possible states of some system, where T specifies how the state changes changes over a specific time interval. For some y X, we define the orbit of y under T to be the sequence y, T (y), T 2(y); , T n(2y); . If T is one-to-one and onto, then we say T is an invertible transformation··· . ··· Ergodic theory studies a particular subset of these dynamical systems{those which are measure-preserving. In the paper, we define and present a number of characteristics pertaining to these measure-preserving dynamical systems, such as randomness and recurrence. We assume a background in basic notions of measure theory and Lebesgue integration. The relevant background information can be found in most real analysis textbooks, such as Real and Complex Analysis by Walter Rudin. The reference we derive the conventions and notations in the subsequent section from is An Invitation to Ergodic Theory by C.E. Silva. Date: DEADLINES: Draft AUGUST 17 and Final version AUGUST 28, 2015. 1 2 WEIAN WANG 1. Recurrence and Ergodicity in Dynamical Systems Suppose we have a nonempty set X and a σ-algebra in X.A measure space is defined to be a triple (X; ; µ) where µ is a measureB on . We say a set A is a measurable set if A X andB A .A probability spaceB is a measure space (X; ; µ) such that µ(X⊂) = 1. We say2 B a measure space is a finite measure space if µ(BX) < and is σ-finite if there exists a sequence of measurable sets A of 1 n finite measure such that X = 1 A . [n=1 n Definition 1.1. Let (X; ; µ) be a probability space. We say the transformation T : X X is measure-preservingB (with respect to µ) and that µ is T -invariant !1 if µ(T − (A)) = µ(A) for all A . 2 B If T is measure-preserving, then we refer to the dynamical system (X; ; µ, T ) as a measure-preserving dynamical system. B We take a measurable set A of positive measure in and consider the orbit of points in A. Specifically, we ask whether the points inBA will return to the set A and if so, how often will they return. We say a measure-preserving transformation T defined on a measure space (X; ; µ) is recurrent if for every measurable set A of positive measure, there is a nullB set N A such that for all x A N , there exists an integer n = n(x) > 0 with T n(x⊂) A. In order words, if,2 forn every measurable set A of positive measure, every point2 in that set, except points in a set of measure zero, eventually returns to A under T , then T is recurrent, and the system is a recurrent dynamical system. The following theorem is a theorem due to Poincar´ewho proved a property relating to finite measure spaces. The statement and full proof of this theorem can be found in [5]. Theorem 1.2 (Poincar´eRecurrence Theorem). Let (X; ; µ) be a finite measure space. If T : X X is a measure-preserving transformation,B then T is recurrent. ! We define a point which is not recurrent to be a wandering point. Formally, a wandering point is a point in A such that n N, T n(x) A. By Poincar´e recurrence, for a finite measure space with a measure-preserving8 2 62 transformation T , the wandering set (i.e. the set of all wandering points in this system) is a set of measure zero. Many of the properties we study that hold for recurrent and ergodic systems hold outside of a set of measure zero; we characterize these properties as holding almost everywhere. To generalize this notion, we define an invariant set. We say 1 a set A is positively invariant if A T − (A) and strictly invariant or simply 1 ⊂ T -invariant if A = T − (A). Studying ergodic properties on T -invariant sets allows us to disregard sets of measure zero. Recurrent transformations on a finite measure space are said to be ergodic if they satisfy the following property. Definition 1.3. A measure-preserving transformation T is ergodic if whenever A is a strictly T -invariant measurable set, then either µ(A) = 0 or µ(Ac) = 0. The following lemma relates our notions of recurrence and ergodicity, and intro- duces some new properties of such systems. Lemma 1.4. Let (X; ; µ) be a σ-finite measure space and let T be a measure- preserving transformation.B Then the following are equivalent: APPLICATIONS OF THE BIRKHOFF ERGODIC THEOREM 3 (1) T is recurrent and ergodic. n (2) For every measurable set A of positive measure, µ(X n1=1 T − (A)) = 0. (3) For every measurable set A of positive measure and forn a.e. x X there exists an integer n > 0 such that T n(x) A. S 2 (4) If A and B are sets of positive measure,2 then there exists an integer n > 0 n such that T − (A) B = . (5) If A and B are sets\ of6 positive; measure, then there exists an integer n > 0 n such that µ(T − (A) B) > 0. \ 1.1. Approximation with Sufficient Semi-rings. While we assume a back- ground in fundamental notions of measure theory, we develop here some techniques of approximation with semi-ring structures. We define a semi-ring to be a collection of subsets of a nonempty space X such that R (1) is nonempty. (2) ifRA; B; , then A B , and (3) if A; B 2 R, then \ 2 R 2 R n A B = E n j j=1 G where E are disjoint. j 2 R We say a semi-ring is a sufficient semi-ring if it satisfies that for every measurable set A in theRσ-algebra, 1 1 µ(A) = inf µ(I ): A I and I for j 1 8 j ⊂ j j 2 R ≥ 9 j=1 j=1 <X [ = There are several useful: properties for studying these structures.; In particular, we have that if a measurable set can be written as a countable union of elements of a semi-ring , then it can be written as a countable union of disjoint elements of the semi-ring.1C We also have the property that every finite measurable set can be approximated, for every > 0, by a finite union of disjoint elements of a sufficient semi-ring where the symmetric difference between the finite measurable set and the sufficient semi-ring is less than . In this examination of ergodic dynamical systems, particularly relevant sufficient semi-rings are the intervals and the dyadic intervals, k k+1 n or the set of all intervals of the form [ n ; n ) where n > 0 and k = 0; 1; ; 2 1. 2 2 ··· − Lemma 1.5. Let (X; ; µ) be a measure space with a sufficient semi-ring . Let B C A be a measurable set, µ(A) < , and let > 0. Then there exists a set H∗ that 1 2 3 is a finite union of disjoint elements of such that µ(A∆H∗ ) < . C This lemma implies that every element of a sufficient semi-ring is a measurable set. When proving properties of measure-preserving systems, it is sufficient to prove them only for the measurable sets which are elements of a sufficient semi-ring in order to verify the properties for all measurable sets in a σ-algebra, i.e. 1This statement is Proposition 2.7.1 in [5]. 2We use ∆ to denote symmetric difference. 3The lemma is a statement of the first of Littlewood's Three Principles of Real Analysis. Its statement and full proof can be found in [5]. 4 WEIAN WANG Theorem 1.6. Let (X; ; µ) be a σ-finite measure space with a sufficient semi-ring . If for all I , B C 1 2 C (1) T − (I) is a measurable set, and 1 (2) µ(T − (I)) = µ(I) then T is a measure-preserving transformation. 2. The Birkhoff Ergodic Theorem The Ergodic Theorem, due to Birkhoff in 1931, relates the time-average of a transformation and the measure of the space. By time-average, we refer to the limit of the average number of times the elements of the sequence x; T (x);T 2(x); are in A, or the average number of times x visits A in its orbit.
Load more

Recommended publications
  • The Generalized Recurrent Set, Explosions and Lyapunov Functions
    The generalized recurrent set, explosions and Lyapunov functions OLGA BERNARDI 1 ANNA FLORIO 2 JIM WISEMAN 3 1 Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Italy 2 Laboratoire de Mathématiques d’Avignon, Avignon Université, France 3 Department of Mathematics, Agnes Scott College, Decatur, Georgia, USA Abstract We consider explosions in the generalized recurrent set for homeomorphisms on a compact metric space. We provide multiple examples to show that such explosions can occur, in contrast to the case for the chain recurrent set. We give sufficient conditions to avoid explosions and discuss their necessity. Moreover, we explain the relations between explosions and cycles for the generalized re- current set. In particular, for a compact topological manifold with dimension greater or equal 2, we characterize explosion phenomena in terms of existence of cycles. We apply our results to give sufficient conditions for stability, under C 0 perturbations, of the property of admitting a continuous Lyapunov function which is not a first integral. 1 Introduction Generalized recurrence was originally introduced for flows by Auslander in the Sixties [3] by using con- tinuous Lyapunov functions. Auslander defined the generalized recurrent set to be the union of those orbits along which all continuous Lyapunov functions are constant. In the same paper, he gave a char- acterization of this set in terms of the theory of prolongations. The generalized recurrent set was later extended to maps by Akin and Auslander (see [1] and [2]). More recently Fathi and Pageault [7] proved that, for a homeomorphism, the generalized recurrent set can be equivalently defined by using Easton’s strong chain recurrence [6].
    [Show full text]
  • Hopf Decomposition and Horospheric Limit Sets
    The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria Hopf Decomposition and Horospheric Limit Sets Vadim A. Kaimanovich Vienna, Preprint ESI 2017 (2008) March 31, 2008 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via http://www.esi.ac.at HOPF DECOMPOSITION AND HOROSPHERIC LIMIT SETS VADIM A. KAIMANOVICH Abstract. By looking at the relationship between the recurrence properties of a count- able group action with a quasi-invariant measure and the structure of its ergodic compo- nents we establish a simple general description of the Hopf decomposition of the action into the conservative and the dissipative parts in terms of the Radon–Nikodym deriva- tives of the action. As an application we prove that the conservative part of the boundary action of a discrete group of isometries of a Gromov hyperbolic space with respect to any invariant quasi-conformal stream coincides (mod 0) with the big horospheric limit set of the group. Conservativity and dissipativity are, alongside with ergodicity, the most basic notions of the ergodic theory and go back to its mechanical and thermodynamical origins. The famous Poincar´erecurrence theorem states that any invertible transformation T preserv- ing a probability measure m on a state space X is conservative in the sense that any positive measure subset A ⊂ X is recurrent, i.e., for a.e. starting point x ∈ A the trajec- tory {T nx} eventually returns to A. These definitions obviously extend to an arbitrary measure class preserving action G (X,m) of a general countable group G on a prob- ability space (X,m).
    [Show full text]
  • ABSTRACT the Specification Property and Chaos In
    ABSTRACT The Specification Property and Chaos in Multidimensional Shift Spaces and General Compact Metric Spaces Reeve Hunter, Ph.D. Advisor: Brian E. Raines, D.Phil. Rufus Bowen introduced the specification property for maps on a compact met- ric space. In this dissertation, we consider some implications of the specification d property for Zd-actions on subshifts of ΣZ as well as on a general compact metric space. In particular, we show that if σ : X X is a continuous Zd-action with ! d a weak form of the specification property on a d-dimensional subshift of ΣZ , then σ exhibits both !-chaos, introduced by Li, and uniform distributional chaos, intro- duced by Schweizer and Smítal. The !-chaos result is further generalized for some broader, directional notions of limit sets and general compact metric spaces with uniform expansion at a fixed point. The Specification Property and Chaos in Multidimensional Shift Spaces and General Compact Metric Spaces by Reeve Hunter, B.A. A Dissertation Approved by the Department of Mathematics Lance L. Littlejohn, Ph.D., Chairperson Submitted to the Graduate Faculty of Baylor University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Approved by the Dissertation Committee Brian E. Raines, D.Phil., Chairperson Nathan Alleman, Ph.D. Will Brian, D.Phil. Markus Hunziker, Ph.D. David Ryden, Ph.D. Accepted by the Graduate School August 2016 J. Larry Lyon, Ph.D., Dean Page bearing signatures is kept on file in the Graduate School. Copyright c 2016 by Reeve Hunter All rights reserved TABLE OF CONTENTS LIST OF FIGURES vi ACKNOWLEDGMENTS vii DEDICATION viii 1 Introduction 1 2 Preliminaries 4 2.1 Dynamical Systems .
    [Show full text]
  • Dimensions for Recurrence Times : Topological and Dynamical Properties
    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 5, Number 4, October 1998 pp. 783{798 DIMENSIONS FOR RECURRENCE TIMES : TOPOLOGICAL AND DYNAMICAL PROPERTIES. VINCENT PENNE¶, BENO^IT SAUSSOL, SANDRO VAIENTI Centre de Physique Th¶eorique, CNRS Luminy, Case 907, F-13288 Marseille - Cedex 9, FRANCE, and PhyMat - D¶epartement de math¶ematique, Universit¶ede Toulon et du Var, 83957 La Garde, FRANCE. Abstract. In this paper we give new properties of the dimension introduced by Afraimovich to characterize Poincar¶erecurrence and which we proposed to call Afraimovich-Pesin's (AP's) dimension. We will show in particular that AP's dimension is a topological invariant and that it often coincides with the asymptotic distribution of periodic points : deviations from this behavior could suggest that the AP's dimension is sensitive to some \non-typical" points. Introduction. The Carath¶eodory construction (see [15] for a complete presenta- tion and historical accounts), has revealed to be a powerful unifying approach for the understanding of thermodynamical formalism and fractal properties of dynam- ically de¯ned sets. A new application of this method has recently been proposed by Afraimovich [1] to characterize Poincar¶erecurrence : it basically consists in the construction of an Hausdor®-like outer measure (with the related transition point, or dimension), but with a few important di®erences. The classical Hausdor® mea- sure (see for instance [8]) is constructed by covering a given set A with arbitrary subsets and by taking the diameter of these subsets at a power ® to build up the Carath¶eodory sum. In Afraimovich's setting, the diameter is replaced with a de- creasing function (gauge function) of the smallest ¯rst return time of the points of each set of the covering into the set itself.
    [Show full text]
  • Chapter 4 Global Behavior: Simple Examples
    Chapter 4 Global Behavior: simple examples Different local behaviors have been analyzed in the previous chapter. Unfortunately, such analysis is insufficient if one wants to understand the global behavior of a Dynamical System. To make precise what we mean by global behavior we need some definitions. Definition 4.0.2 Given a Dynamical System (X; φt), t 2 N or R+, a −1 set A ⊂ X is called invariant if, for all t, ; 6= φt (A) ⊂ A. Essentially, the global understanding of a system entails a detailed knowledge of its invariant set and of the dynamics in a neighborhood of such sets. This is in general very hard to achieve, essentially the rest of this book devoted to the study of some special cases. Remark 4.0.3 We start with some simple considerations in the case of continuous Dynamical Systems (this is part of a general theory called Topological Dynamical Systems1) and then we will address more subtle phenomena that depend on the smoothness of the systems. 4.1 Long time behavior and invariant sets First of all let us note that if we are interested in the long time behavior of a system and we look at it locally (i.e. in the neighborhood of a point) then three cases are possible: either the motion leaves the neighborhood 1 Recall that a Topological Dynamical Systems is a couple (X; φt) where X is a topological space and φt is a continuous action of R (or R+; N; Z) on X. 71 72 CHAPTER 4. GLOBAL BEHAVIOR: SIMPLE EXAMPLES and never returns, or leaves the neighborhood but eventually it comes back or never leaves.
    [Show full text]
  • Distinguishability Notion Based on Wootters Statistical Distance: Application to Discrete Maps
    Distinguishability notion based on Wootters statistical distance: application to discrete maps Ignacio S. Gomez,1, ∗ M. Portesi,1, y and P. W. Lamberti2, z 1IFLP, UNLP, CONICET, Facultad de Ciencias Exactas, Calle 115 y 49, 1900 La Plata, Argentina 2Facultad de Matem´atica, Astronom´ıa y F´ısica (FaMAF), Universidad Nacional de C´ordoba, Avenida Medina Allende S/N, Ciudad Universitatia, X5000HUA, C´ordoba, Argentina (Dated: August 10, 2018) We study the distinguishability notion given by Wootters for states represented by probability density functions. This presents the particularity that it can also be used for defining a distance in chaotic unidimensional maps. Based on that definition, we provide a metric d for an arbitrary discrete map. Moreover, from d we associate a metric space to each invariant density of a given map, which results to be the set of all distinguished points when the number of iterations of the map tends to infinity. Also, we give a characterization of the wandering set of a map in terms of the metric d which allows to identify the dissipative regions in the phase space. We illustrate the results in the case of the logistic and the circle maps numerically and theoretically, and we obtain d and the wandering set for some characteristic values of their parameters. PACS numbers: 05.45.Ac, 02.50.Cw, 0.250.-r, 05.90.+m Keywords: discrete maps { invariant density { metric erarchy [11], with quantum extensions [12{16] that allow space { wandering set to characterize aspects of quantum chaos [17]. The rele- vance of discrete maps lies in the fact that they serve as simple but useful models in biology, physics, economics, I.
    [Show full text]
  • Arxiv:2009.13713V2 [Math.DS]
    CHAOS AND FREQUENT HYPERCYCLICITY FOR COMPOSITION OPERATORS UDAYAN B. DARJI AND BENITO PIRES Abstract. The notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Ruzsa in 2015 that for backward weighted shifts on ℓp(Z), the notions chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on c0. It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on Lp-spaces the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible frequently hypercyclic operator on ℓ1 whose inverse is not frequently hypercyclic is constructed. 1. Introduction Let H be a separable Banach space and T : H → H a continuous linear operator. T is chaotic if T has a dense orbit and the set of periodic points of T is dense in X. Chaotic operators are well studied in linear dynamics and well understood. For example, Grosse-Erdmann in [12] gave a complete characterization of backward weighted shifts which are chaotic. The notion of frequent hypercyclicity is a quantitative version of hypercyclicity. It was introduced by Bayart and Grivaux in [3] in the linear setting, but it also makes sense for Polish dynamical systems.
    [Show full text]
  • Distinguishability Notion Based on Wootters Statistical Distance: Application to Discrete Maps Ignacio S
    Distinguishability notion based on Wootters statistical distance: Application to discrete maps Ignacio S. Gomez, M. Portesi, and P. W. Lamberti Citation: Chaos 27, 083112 (2017); doi: 10.1063/1.4998141 View online: https://doi.org/10.1063/1.4998141 View Table of Contents: http://aip.scitation.org/toc/cha/27/8 Published by the American Institute of Physics Articles you may be interested in Ag fractal structures in electroless metal deposition systems with and without magnetic field Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 083111 (2017); 10.1063/1.4997762 Analysis and application of opinion model with multiple topic interactions Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 083113 (2017); 10.1063/1.4998736 Large deviation induced phase switch in an inertial majority-vote model Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 081102 (2017); 10.1063/1.4993520 Amplitude death in a ring of nonidentical nonlinear oscillators with unidirectional coupling Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 083119 (2017); 10.1063/1.4999910 Constructing an autonomous system with infinitely many chaotic attractors Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 071101 (2017); 10.1063/1.4986356 Detection of coupling delay: A problem not yet solved Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 083109 (2017); 10.1063/1.4997757 CHAOS 27, 083112 (2017) Distinguishability notion based on Wootters statistical distance: Application to discrete maps Ignacio S. Gomez,1,a) M. Portesi,1,b) and P. W. Lamberti2,c) 1IFLP, UNLP, CONICET, Facultad de Ciencias Exactas, Calle 115 y 49, 1900 La Plata, Argentina 2Facultad de Matematica, Astronomıa, Fısica y Computacion (FaMAF), Universidad Nacional de Cordoba, Avenida Medina Allende S/N, Ciudad Universitatia, X5000HUA Cordoba, Argentina (Received 18 March 2017; accepted 28 July 2017; published online 15 August 2017) We study the distinguishability notion given by Wootters for states represented by probability density functions.
    [Show full text]
  • 10. the Ergodic Theory of Hyperbolic Dynamical Systems
    MAGIC010 Ergodic Theory Lecture 10 10. The ergodic theory of hyperbolic dynamical systems §10.1 Introduction In Lecture 8 we studied thermodynamic formalism for shifts of finite type by defining a suitable transfer operator acting on a certain Banach space of functions and studying its spectral properties. In Lecture 9 we gave some applications of this methodology, mostly in the context of shifts of finite type. In this lecture we show how the use of thermodynamic formalism can be used to study a wide range of dynamical system that possesses some degree of ‘hyperbolicity’. §10.2 Anosov diffeomorphisms Let M be a compact Riemannian manifold without boundary, with metric d on M derived from the Riemannian metric. For x ∈ M let TxM denote the tangent space at x and let TM denote the tangent bundle. Let T : M → M 1 be a C diffeomorphism of M and let DxT : TxM → TT (x)M denote the derivative of T . Definition. We say that T : M → M is an Anosov diffeomorphism if the tangent bundle TM has a continuous splitting into a Whitney sum of two DT -invariant sub-bundles TM = Es ⊕ Eu such that there exist constant C > 0 and λ ∈ (0, 1) such that for all n ≥ 0 n n s kDxT vk ≤ Cλ kvk, for all v ∈ Ex, −n n u kDxT vk ≤ Cλ kvk, for all v ∈ Ex . We call Es and Eu the stable and unstable sub-bundles, respectively. Thus T is an Anosov diffeomorphism if each tangent plane splits into two complementary directions; one of which contracts exponentially fast under iteration and the other expands exponentially fast under iteration.
    [Show full text]
  • PERSISTENT MASSIVE ATTRACTORS of SMOOTH MAPS 3 Is Already Enough to Create Robust T-Big Attractors
    PERSISTENT MASSIVE ATTRACTORS OF SMOOTH MAPS D. VOLK Abstract. For a smooth manifold of any dimension greater than one, we present an open set of smooth endomorphisms such that any of them has a transitive attractor with a non-empty interior. These maps are m-fold non-branched coverings, m ≥ 3. The construction applies to any manifold of the form S1 ×M, where S1 is the standard circle and M is an arbitrary manifold. 1. Introduction Consider a smooth map F of a compact manifold X (possibly with boundary) into itself. The semigroup generated by F is a discrete-time dynamical system. In the theory of dynamical systems, we are interested in the limit behavior of the orbits. In a wide class of dynamical systems, one can find a proper subset Λ ⊂ X and an open set B ⊃ Λ such that the trajectories of almost every point in B accumulate (in some sense) onto Λ. Then Λ is said to be an attractor and B its basin. Various formalizations of the word “accumulate” give rise to different (usually non-equivalent) definitions of attractors. The proper identification of an attractor is very important in applications because it is often possible to reduce the dimensions or size of the system by restricting it to its attractor. We do not consider the trivial case, Λ = X. In this situation, the notion of an attractor does not provide any additional information on the system. Now we address the following natural question suggested by Yu. Ilyashenko: if Λ =6 X, how big can Λ be? There are many ways to tell what is “big”.
    [Show full text]
  • Genericity of Geodesic Flows with Positive Topological Entropy on S2
    GENERICITY OF GEODESIC FLOWS WITH POSITIVE TOPOLOGICAL ENTROPY ON S2 GONZALO CONTRERAS-BARANDIARAN´ AND GABRIEL P. PATERNAIN Abstract. We show that the set of C∞ riemannian metrics on S2 or RP 2 whose geodesic flow has positive topological entropy is open and dense in the C2 topology. The proof is partially based on an analogue of Franks’ lemma for geodesic flows on surfaces. To the memory of Michel Herman 1. Introduction ∞ g Let M be a closed surface endowed with a C riemannian metric g and let φt be the geodesic flow of g. One of the most important dynamical invariants that one can associate g to φt to roughly measure its orbit structure complexity is the topological entropy, which we shall denote by htop(g). The first question one asks about htop(g) is whether it vanishes g or not. If htop(g) > 0 a well known result of A. Katok [25] states that the dynamics of φt presents transverse homoclinic intersections and as a consequence the number of periodic hyperbolic geodesics grows exponentially with length. Moreover, other conclusions of a more geometrical nature can be drawn. Given p and q in M and T > 0, define nT (p, q) as the number of geodesic arcs joining p and q with length ≤ T . R. Ma˜n´eshowed in [35] that Z 1 lim log nT (p, q) dp dq = htop(g), T →∞ T M×M and therefore if htop(g) > 0, we have that on average the number of arcs between two points grows exponentially with length. Even better, K.
    [Show full text]
  • Hopf Decomposition and Horospheric Limit Sets
    Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 35, 2010, 335–350 HOPF DECOMPOSITION AND HOROSPHERIC LIMIT SETS Vadim A. Kaimanovich University of Ottawa, Department of Mathematics and Statistics 585 King Edward Ave, Ottawa ON, K1N 6N5, Canada; [email protected] Abstract. By looking at the relationship between the recurrence properties of a countable group action with a quasi-invariant measure and the structure of the space of ergodic components, we establish a simple general description of the Hopf decomposition of the action into its conservative and dissipative parts in terms of the Radon–Nikodym derivatives. As an application we describe the conservative part of the boundary action of a discrete group of isometries of a Gromov hyperbolic space with respect to an invariant quasi-conformal stream. Conservativity and dissipativity are, along with ergodicity, the most basic notions of ergodic theory which go back to its mechanical and thermodynamical origins. The famous Poincaré recurrence theorem states that any measure preserving transforma- tion T of a probability space (X; m) is conservative in the sense that any positive measure subset A ½ X is recurrent, i.e., for a.e. starting point x 2 A the trajectory fT nxg eventually returns to A. These definitions clearly make sense for any measure class preserving action G ª (X; m) of a general countable group G on a probability space. The opposite notions are those of dissipativity and of a wandering set, i.e., such a set A that all its translates gA; g 2 G; are pairwise disjoint. An action is called dissipative if it admits a wandering set of positive measure, and it is called completely dissipative if, moreover, there is a wandering set such that the union of its translates is (mod 0) the whole action space.
    [Show full text]