Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms
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Anosov Diffeomorphism with a Horseshoe That Attracts Almost Any Point
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2020017 DYNAMICAL SYSTEMS Volume 40, Number 1, January 2020 pp. 441–465 ANOSOV DIFFEOMORPHISM WITH A HORSESHOE THAT ATTRACTS ALMOST ANY POINT Christian Bonatti Institut de Math´ematiquesde Bourgogne, Universit´ede Bourgogne Dijon 21004, France Stanislav Minkov Brook Institute of Electronic Control Machines 119334, Moscow, Vavilova str., 24, Russia Alexey Okunev Loughborough University LE11 3TU, Loughborough, UK Ivan Shilin National Research University Higher School of Economics Faculty of Mathematics, 119048, Moscow, Usacheva str., 6, Russia (Communicated by Sylvain Crovisier) Abstract. We present an example of a C1 Anosov diffeomorphism of a two- torus with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure. 1. Introduction. Consider a diffeomorphism F that preserves a probability mea- sure ν. The basin of ν is the set of all points x such that the sequence of measures n 1 δ := (δ + ··· + δ n−1 ) x n x F (x) converges to ν in the weak-∗ topology. A measure is called physical if its basin has positive Lebesgue measure. It is well known (see [3], [11], [12], or [1, x1:3]) that any transitive C2 Anosov diffeomorphism (note that all known Anosov diffeomorphisms are transitive) has a unique physical measure and • the basin of this measure has full Lebesgue measure, • the support of this measure coincides with the whole phase space. In C1 dynamics there are many phenomena that are impossible in C2. Bowen [2] has constructed an example of a C1 diffeomorphism of the plane with a thick horseshoe (i.e., a horseshoe with positive Lebesgue measure) and Robinson and Young [10] have embedded a thick horseshoe in a C1 Anosov diffeomorphism of T2 2010 Mathematics Subject Classification. -
Invariant Measures for Hyperbolic Dynamical Systems
Invariant measures for hyperbolic dynamical systems N. Chernov September 14, 2006 Contents 1 Markov partitions 3 2 Gibbs Measures 17 2.1 Gibbs states . 17 2.2 Gibbs measures . 33 2.3 Properties of Gibbs measures . 47 3 Sinai-Ruelle-Bowen measures 56 4 Gibbs measures for Anosov and Axiom A flows 67 5 Volume compression 86 6 SRB measures for general diffeomorphisms 92 This survey is devoted to smooth dynamical systems, which are in our context diffeomorphisms and smooth flows on compact manifolds. We call a flow or a diffeomorphism hyperbolic if all of its Lyapunov exponents are different from zero (except the one in the flow direction, which has to be 1 zero). This means that the tangent vectors asymptotically expand or con- tract exponentially fast in time. For many reasons, it is convenient to assume more than just asymptotic expansion or contraction, namely that the expan- sion and contraction of tangent vectors happens uniformly in time. Such hyperbolic systems are said to be uniformly hyperbolic. Historically, uniformly hyperbolic flows and diffeomorphisms were stud- ied as early as in mid-sixties: it was done by D. Anosov [2] and S. Smale [77], who introduced his Axiom A. In the seventies, Anosov and Axiom A dif- feomorphisms and flows attracted much attention from different directions: physics, topology, and geometry. This actually started in 1968 when Ya. Sinai constructed Markov partitions [74, 75] that allowed a symbolic representa- tion of the dynamics, which matched the existing lattice models in statistical mechanics. As a result, the theory of Gibbs measures for one-dimensional lat- tices was carried over to Anosov and Axiom A dynamical systems. -
Linear Response, Or Else
Linear response, or else Viviane Baladi Abstract. Consider a smooth one-parameter family t ft of dynamical systems ft, with t < ϵ. !→ | | Assume that for all t (or for many t close to t =0) the map ft admits a unique physical invariant probability measure µt. We say that linear response holds if t µt is differentiable at t =0(possibly !→ in the sense of Whitney), and if its derivative can be expressed as a function of f0, µ0, and ∂tft t=0. | The goal of this note is to present to a general mathematical audience recent results and open problems in the theory of linear response for chaotic dynamical systems, possibly with bifurcations. Mathematics Subject Classification (2010). Primary 37C40; Secondary 37D25, 37C30, 37E05. Keywords. Linear response, transfer operator, Ruelle operator, physical measure, SRB measure, bi- furcations, differentiable dynamical system, unimodal maps, expanding interval maps, hyperbolic dy- namical systems. 1. Introduction A discrete-time dynamical system is a self-map f : M M on a space M. To any point n → 0 x M is then associated its (future) orbit f (x) n Z+ where f (x)=x, and n∈ n 1 { | ∈ } f (x)=f − (f(x)), for n 1, represents the state of the system at time n, given the “ini- ≥ n tial condition” x. (If f is invertible, one can also consider the past orbit f − (x) n Z+ .) { | ∈ } In this text, we shall always assume that M is a compact differentiable manifold (possibly with boundary), with the Borel σ-algebra, endowed with a Riemannian structure and thus normalised Lebesgue measure. -
Arxiv:2003.06811V3 [Math.DS] 31 Mar 2021 Nsmcasclaayi Eg E [ See (E.G
ANOSOV DIFFEOMORPHISMS, ANISOTROPIC BV SPACES AND REGULARITY OF FOLIATIONS WAEL BAHSOUN AND CARLANGELO LIVERANI Abstract. Given any smooth Anosov map we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities. 1. Introduction Starting with the paper [BKL], there has been a growing interest in the pos- sibility to develop a functional analytic setting allowing the direct study of the transfer operator of a hyperbolic dynamical system. The papers [GL, GL1, BT, BT1, B1, B2, B3, B4, T1] have now produced quite satisfactory results for the case of Anosov diffeomorphisms (or, more generally, for uniformly hyperbolic basic sets). Important results, although the theory is not complete yet, have been ob- tained for flows [L, BuL, BuL2, GLP, FT2, DyZ, D17], group extensions and skew products ([F11, AGT]). Moreover, recently a strong relation with techniques used in semiclassical analysis (e.g. see [FR, FRS, FT1, FT2, DyZ]) has been unveiled. Also, one should mention the recent discovery of a deep relation with the theory of renormalization of parabolic systems [GL19]. In addition, such an approach has proven very effective in the study of perturbation of dynamical systems [KL1, KL3] and in the investigation of limit theorems [G10]. -
What Are Lyapunov Exponents, and Why Are They Interesting?
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 1, January 2017, Pages 79–105 http://dx.doi.org/10.1090/bull/1552 Article electronically published on September 6, 2016 WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING? AMIE WILKINSON Introduction At the 2014 International Congress of Mathematicians in Seoul, South Korea, Franco-Brazilian mathematician Artur Avila was awarded the Fields Medal for “his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.”1 Although it is not explicitly mentioned in this citation, there is a second unify- ing concept in Avila’s work that is closely tied with renormalization: Lyapunov (or characteristic) exponents. Lyapunov exponents play a key role in three areas of Avila’s research: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schr¨odinger operators. Here we take the op- portunity to explore these areas and reveal some underlying themes connecting exponents, chaotic dynamics and renormalization. But first, what are Lyapunov exponents? Let’s begin by viewing them in one of their natural habitats: the iterated barycentric subdivision of a triangle. When the midpoint of each side of a triangle is connected to its opposite vertex by a line segment, the three resulting segments meet in a point in the interior of the triangle. The barycentric subdivision of a triangle is the collection of 6 smaller triangles determined by these segments and the edges of the original triangle: Figure 1. Barycentric subdivision. Received by the editors August 2, 2016. -
The Ergodic Theory of Hyperbolic Dynamical Systems
MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013 In this lecture we use symbolic dynamics to model more general hyperbolic dynamical systems. We can then use thermodynamic formalism to prove ergodic-theoretic results about such systems. In the last two lectures we studied thermodynamic formalism in the context of one-sided aperiodic shifts of finite type. In the last two lectures we studied thermodynamic formalism in the context of one-sided aperiodic shifts of finite type. In this lecture we use symbolic dynamics to model more general hyperbolic dynamical systems. We can then use thermodynamic formalism to prove ergodic-theoretic results about such systems. Let Tx M be the tangent space at x 2 M. Let Dx T : Tx M ! TTx M be the derivative of T . Idea: At each point x 2 M, there are two directions: I T contracts exponentially fast in one direction. I T expands exponentially fast in the other. Definition T : M ! M is an Anosov diffeomorphism if 9C > 0; λ 2 (0; 1) s.t. 8x 2 M, there is a splitting s u Tx M = Ex ⊕ Ex into DT -invariant sub-bundles E s , E u s.t. n n s kDx T (v)k ≤ Cλ kvk 8v 2 Ex ; 8n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk 8v 2 Ex ; 8n ≥ 0: Hyperbolic dynamical systems Let T be a C 1 diffeomorphism of a compact Riemannian manifold M. Let Dx T : Tx M ! TTx M be the derivative of T . -
Uniformly Hyperbolic Control Theory Christoph Kawan
HYPERBOLIC CONTROL THEORY 1 Uniformly hyperbolic control theory Christoph Kawan Abstract—This paper gives a summary of a body of work at the of control-affine systems with a compact and convex control intersection of control theory and smooth nonlinear dynamics. range. The main idea is to transfer the concept of uniform hyperbolicity, These results are grounded on the topological theory of central to the theory of smooth dynamical systems, to control- affine systems. Combining the strength of geometric control Colonius-Kliemann [7] which provides an approach to under- theory and the hyperbolic theory of dynamical systems, it is standing the global controllability structure of control systems. possible to deduce control-theoretic results of non-local nature Two central notions of this theory are control and chain that reveal remarkable analogies to the classical hyperbolic the- control sets. Control sets are the maximal regions of complete ory of dynamical systems. This includes results on controllability, approximate controllability in the state space. The definition of robustness, and practical stabilizability in a networked control framework. chain control sets involves the concept of "-chains (also called "-pseudo-orbits) from the theory of dynamical systems. The Index Terms—Control-affine system; uniform hyperbolicity; main motivation for this concept comes from the facts that (i) chain control set; controllability; robustness; networked control; invariance entropy chain control sets are outer approximations of control sets and (ii) chain control sets in general are easier to determine than control sets (both analytically and numerically). I. INTRODUCTION As examples show, chain control sets can support uniformly hyperbolic and, more generally, partially hyperbolic structures. -
Maximal Entropy Measures for Certain Partially Hyperbolic, Derived from Anosov Systems
MAXIMAL ENTROPY MEASURES FOR CERTAIN PARTIALLY HYPERBOLIC, DERIVED FROM ANOSOV SYSTEMS J. BUZZI, T. FISHER, M. SAMBARINO, C. VASQUEZ´ Abstract. We show that a class of robustly transitive diffeomor- phisms originally described by Ma˜n´eare intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which peri- odic orbits are equidistributed. Moreover, equipped with their re- spective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e., produced by an iso- topy from an Anosov system, namely, a mixed Ma˜n´eexample and one obtained through a Hopf bifurcation. 1. Introduction Let f be a diffeomorphism of a manifold M to itself. The diffeomor- phism f is transitive if there exists a point x ∈ M where + n Of (x)= {f (x)|n ∈ N} is dense in M. It is robustly transitive [4, Ch. 7] if there exists a neighborhood U of f in the space Diff1(M) of C1 diffeomorphisms such that each g in U is transitive. Since robust transitivity is an open condition, it is an important component of the global picture of dynamical systems [23]. Date: October 11, 2010. 2000 Mathematics Subject Classification. 37C40, 37A35, 37C15. Key words and phrases. Measures of maximal entropy, topological entropy, ro- bust ergodicity, ergodic theory, partial hyperbolicity, intrinsic ergodicity. -
Hyperbolic Dynamical Systems, Chaos, and Smale's
Hyperbolic dynamical systems, chaos, and Smale's horseshoe: a guided tour∗ Yuxi Liu Abstract This document gives an illustrated overview of several areas in dynamical systems, at the level of beginning graduate students in mathematics. We start with Poincar´e's discovery of chaos in the three-body problem, and define the Poincar´esection method for studying dynamical systems. Then we discuss the long-term behavior of iterating a n diffeomorphism in R around a fixed point, and obtain the concept of hyperbolicity. As an example, we prove that Arnold's cat map is hyperbolic. Around hyperbolic fixed points, we discover chaotic homoclinic tangles, from which we extract a source of the chaos: Smale's horseshoe. Then we prove that the behavior of the Smale horseshoe is the same as the binary shift, reducing the problem to symbolic dynamics. We conclude with applications to physics. 1 The three-body problem 1.1 A brief history The first thing Newton did, after proposing his law of gravity, is to calculate the orbits of two mass points, M1;M2 moving under the gravity of each other. It is a basic exercise in mechanics to show that, relative to one of the bodies M1, the orbit of M2 is a conic section (line, ellipse, parabola, or hyperbola). The second thing he did was to calculate the orbit of three mass points, in order to study the sun-earth-moon system. Here immediately he encountered difficulty. And indeed, the three body problem is extremely difficult, and the n-body problem is even more so. -
Entropy Rigidity for 3D Conservative Anosov Flows and Dispersing Billiards
ENTROPY RIGIDITY FOR 3D CONSERVATIVE ANOSOV FLOWS AND DISPERSING BILLIARDS JACOPO DE SIMOI, MARTIN LEGUIL∗, KURT VINHAGE, AND YUN YANG Abstract. Given an integer k ≥ 5, and a Ck Anosov flow Φ on some compact connected 3-manifold preserving a smooth volume, we show that the measure of maximal entropy (MME) is the volume measure if and only if Φ is Ck−"- conjugate to an algebraic flow, for " > 0 arbitrarily small. Besides the rigidity, we also study the entropy flexibility, and show that the metric entropy with respect to the volume measure and the topological entropy of suspension flows over Anosov diffeomorphisms on the 2-torus achieve all possible values subject to natural normalizations. Moreover, in the case of dispersing billiards, we show that if the measure of maximal entropy is the volume measure, then the Birkhoff Normal Form of regular periodic orbits with a homoclinic intersection is linear. Contents 1. Introduction2 1.1. Outline of arguments and new techniques3 1.2. Entropy flexibility5 1.3. Entropy rigidity for dispersing billiards5 2. Preliminaries7 2.1. General facts about hyperbolic flows7 2.2. Equilibrium states for Anosov/Axiom A flows8 2.3. Anosov class and Birkhoff Normal Form 10 3. Entropy rigidity for conservative Anosov flows on 3-manifolds 12 3.1. Periodic Lyapunov exponents when SRB=MME 12 3.2. Expansion of the Lyapunov exponents of periodic orbits in a horseshoe with prescribed combinatorics 13 3.3. From smooth orbit equivalence to smooth flow conjugacy 19 4. Entropy Flexibility 21 5. Some results towards entropy rigidity for dispersing billiards 26 5.1. -
Transformations)
TRANSFORMACJE (TRANSFORMATIONS) Transformacje (Transformations) is an interdisciplinary refereed, reviewed journal, published since 1992. The journal is devoted to i.a.: civilizational and cultural transformations, information (knowledge) societies, global problematique, sustainable development, political philosophy and values, future studies. The journal's quasi-paradigm is TRANSFORMATION - as a present stage and form of development of technology, society, culture, civilization, values, mindsets etc. Impacts and potentialities of change and transition need new methodological tools, new visions and innovation for theoretical and practical capacity-building. The journal aims to promote inter-, multi- and transdisci- plinary approach, future orientation and strategic and global thinking. Transformacje (Transformations) are internationally available – since 2012 we have a licence agrement with the global database: EBSCO Publishing (Ipswich, MA, USA) We are listed by INDEX COPERNICUS since 2013 I TRANSFORMACJE(TRANSFORMATIONS) 3-4 (78-79) 2013 ISSN 1230-0292 Reviewed journal Published twice a year (double issues) in Polish and English (separate papers) Editorial Staff: Prof. Lech W. ZACHER, Center of Impact Assessment Studies and Forecasting, Kozminski University, Warsaw, Poland ([email protected]) – Editor-in-Chief Prof. Dora MARINOVA, Sustainability Policy Institute, Curtin University, Perth, Australia ([email protected]) – Deputy Editor-in-Chief Prof. Tadeusz MICZKA, Institute of Cultural and Interdisciplinary Studies, University of Silesia, Katowice, Poland ([email protected]) – Deputy Editor-in-Chief Dr Małgorzata SKÓRZEWSKA-AMBERG, School of Law, Kozminski University, Warsaw, Poland ([email protected]) – Coordinator Dr Alina BETLEJ, Institute of Sociology, John Paul II Catholic University of Lublin, Poland Dr Mirosław GEISE, Institute of Political Sciences, Kazimierz Wielki University, Bydgoszcz, Poland (also statistical editor) Prof. -
Prevalence of Non-Lipschitz Anosov Foliations
ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Pages 93{98 (September 11, 1997) S 1079-6762(97)00030-9 PREVALENCE OF NON-LIPSCHITZ ANOSOV FOLIATIONS BORIS HASSELBLATT AND AMIE WILKINSON (Communicated by Krystyna Kuperberg) To the memory of Gunnar Hasselblatt, 19.8.1928–12.7.1997 Abstract. We give sharp regularity results for the invariant subbundles of hyperbolic dynamical systems and give open dense sets of codimension one systems where this regularity is not exceeded as well as open dense sets of symplectic, geodesic, and codimension one systems where the analogous regu- larity results of Pugh, Shub, and Wilkinson are optimal. We produce open sets of symplectic Anosov diffeomorphisms and flows with low transverse H¨older regularity of the invariant foliations almost everywhere. Prevalence of low reg- ularity of conjugacies on large sets is a corollary. We also establish a new connection between the transverse regularity of foliations and their tangent subbundles. 1. Introduction A diffeomorphism f of a compact Riemannian manifold M is called Anosov if the tangent bundle splits (necessarily uniquely and continuously) into two Df-invariant subbundles TM = Eu Es, and there exist constants C and a<1 such that ⊕ n n n n s u Df (v) Ca v and Df− (u) Ca u for v E ;u E;n N: k k≤ k k k k≤ k k ∈ ∈ ∈ Thus, for two nearby points positive or negative iterates move apart exponentially fast. Any hyperbolic matrix A SL(m; Z) induces an Anosov diffeomorphism of ∈ the m-torus Tm; Eu and Es are the expanding and contracting root spaces of A.