Generic Properties of Dynamical Systems Christian Bonatti
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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. -
Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms
Rufus Bowen Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms New edition of Lect. Notes in Math. 470, Springer, 1975. April 14, 2013 Springer Preface The Greek and Roman gods, supposedly, resented those mortals endowed with superlative gifts and happiness, and punished them. The life and achievements of Rufus Bowen (1947{1978) remind us of this belief of the ancients. When Rufus died unexpectedly, at age thirty-one, from brain hemorrhage, he was a very happy and successful man. He had great charm, that he did not misuse, and superlative mathematical talent. His mathematical legacy is important, and will not be forgotten, but one wonders what he would have achieved if he had lived longer. Bowen chose to be simple rather than brilliant. This was the hard choice, especially in a messy subject like smooth dynamics in which he worked. Simplicity had also been the style of Steve Smale, from whom Bowen learned dynamical systems theory. Rufus Bowen has left us a masterpiece of mathematical exposition: the slim volume Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Springer Lecture Notes in Mathematics 470 (1975)). Here a number of results which were new at the time are presented in such a clear and lucid style that Bowen's monograph immediately became a classic. More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems. The area discussed by Bowen came into existence through the merging of two apparently unrelated theories. -
Category Theory for Autonomous and Networked Dynamical Systems
Discussion Category Theory for Autonomous and Networked Dynamical Systems Jean-Charles Delvenne Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM) and Center for Operations Research and Econometrics (CORE), Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium; [email protected] Received: 7 February 2019; Accepted: 18 March 2019; Published: 20 March 2019 Abstract: In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way. Keywords: ergodic theory; topological dynamics; control theory 1. Introduction The theory of autonomous dynamical systems has developed in different flavours according to which structure is assumed on the state space—with topological dynamics (studying continuous maps on topological spaces) and ergodic theory (studying probability measures invariant under the map) as two prominent examples. These theories have followed parallel tracks and built multiple bridges. For example, both have grown successful tools from the interaction with information theory soon after its emergence, around the key invariants, namely the topological entropy and metric entropy, which are themselves linked by a variational theorem. The situation is more complex and heterogeneous in the field of what we here call open dynamical systems, or controlled dynamical systems. -
Fine Structure of Hyperbolic Diffeomorphisms, by A. A. Pinto, D
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 48, Number 1, January 2011, Pages 131–136 S 0273-0979(2010)01284-2 Article electronically published on May 24, 2010 Fine structure of hyperbolic diffeomorphisms,byA.A.Pinto,D.Rand,andF.Fer- reira, Springer Monographs in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2009, xvi+354 pp., ISBN 978-3-540-87524-6, hardcover, US$129.00 The main theme of the book Fine Structures of Hyperbolic Diffeomorphisms,by Pinto, Rand and Ferreira, is the rigidity and flexibility of two-dimensional diffeo- morphisms on hyperbolic basic sets and properties of invariant measures that are related to the geometry of these invariant sets. In his remarkable article [23], Smale sets the foundations of the modern theory of dynamical systems. He defines the fundamental notion of hyperbolicity and relates it to structural stability. Let f be a smooth (at least C1) diffeomorphism of a compact manifold M. A hyperbolic set for f is a closed f-invariant subset Λ ⊂ M such that the tangent bundle of the manifold over Λ splits as a direct sum of two subbundles that are invariant under the derivative, and the derivative of the iterates of the map expands exponentially one of the bundles (the unstable bundle) and contracts exponentially the stable subbundle. These bundles are in general only continuous, but they are integrable. Through each point x ∈ Λ, there exists a one-to-one immersed submanifold W s(x), the stable manifold of x.This submanifold is tangent to the stable bundle at each point of intersection with Λ and is characterized by the fact that the orbit of each point y ∈ W s(x)isasymptotic to the orbit of x, and, in fact, the distance between f n(y)tof n(x)converges to zero exponentially fast. -
Topological Entropy and Distributional Chaos in Hereditary Shifts with Applications to Spacing Shifts and Beta Shifts
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2013.33.2451 DYNAMICAL SYSTEMS Volume 33, Number 6, June 2013 pp. 2451{2467 TOPOLOGICAL ENTROPY AND DISTRIBUTIONAL CHAOS IN HEREDITARY SHIFTS WITH APPLICATIONS TO SPACING SHIFTS AND BETA SHIFTS Dedicated to the memory of Professor Andrzej Pelczar (1937-2010). Dominik Kwietniak Faculty of Mathematics and Computer Science Jagiellonian University in Krak´ow ul.Lojasiewicza 6, 30-348 Krak´ow,Poland (Communicated by Llu´ısAlsed`a) Abstract. Positive topological entropy and distributional chaos are charac- terized for hereditary shifts. A hereditary shift has positive topological entropy if and only if it is DC2-chaotic (or equivalently, DC3-chaotic) if and only if it is not uniquely ergodic. A hereditary shift is DC1-chaotic if and only if it is not proximal (has more than one minimal set). As every spacing shift and every beta shift is hereditary the results apply to those classes of shifts. Two open problems on topological entropy and distributional chaos of spacing shifts from an article of Banks et al. are solved thanks to this characterization. Moreover, it is shown that a spacing shift ΩP has positive topological entropy if and only if N n P is a set of Poincar´erecurrence. Using a result of Kˇr´ıˇzan example of a proximal spacing shift with positive entropy is constructed. Connections between spacing shifts and difference sets are revealed and the methods of this paper are used to obtain new proofs of some results on difference sets. 1. Introduction. A hereditary shift is a (one-sided) subshift X such that x 2 X and y ≤ x (coordinate-wise) imply y 2 X. -
Topological Dynamics of Ordinary Differential Equations and Kurzweil Equations*
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF DIFFERENTIAL EQUATIONS 23, 224-243 (1977) Topological Dynamics of Ordinary Differential Equations and Kurzweil Equations* ZVI ARTSTEIN Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912 Received March 25, 1975; revised September 4, 1975 1. INTRODUCTION A basic construction in applying topological dynamics to nonautonomous ordinary differential equations f =f(x, s) is to consider limit points (when 1 t 1 + co) of the translatesf, , wheref,(x, s) = f(~, t + s). A limit point defines a limiting equation and there is a tight relation- ship between the behavior of solutions of the original equation and of the limiting equations (see [l&13]). I n order to carry out this construction one has to specify the meaning of the convergence of ft , and this convergence has to have certain properties which we shall discuss later. In this paper we shall encounter a phenomenon which seems to be new in the context of applications of topological dynamics. The limiting equations will not be ordinary differential equations. We shall work under assumptions that do not guarantee that the space of ordinary equations is complete and we shall have to consider a completion of the space. Almost two decades ago Kurzweil [3] introduced what he called generalized ordinary differential equations and what we call Kurzweil equations. We shall see that if we embed our ordinary equations in a space of Kurzweil equations we get a complete and compact space, and the techniques of topological dynamics can be applied. -
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. -
Groups and Topological Dynamics Volodymyr Nekrashevych
Groups and topological dynamics Volodymyr Nekrashevych E-mail address: [email protected] 2010 Mathematics Subject Classification. Primary ... ; Secondary ... Key words and phrases. .... The author ... Abstract. ... Contents Chapter 1. Dynamical systems 1 x1.1. Introduction by examples 1 x1.2. Subshifts 21 x1.3. Minimal Cantor systems 36 x1.4. Hyperbolic dynamics 74 x1.5. Holomorphic dynamics 103 Exercises 111 Chapter 2. Group actions 117 x2.1. Structure of orbits 117 x2.2. Localizable actions and Rubin's theorem 136 x2.3. Automata 149 x2.4. Groups acting on rooted trees 163 Exercises 193 Chapter 3. Groupoids 201 x3.1. Basic definitions 201 x3.2. Actions and correspondences 214 x3.3. Fundamental groups 233 x3.4. Orbispaces and complexes of groups 238 x3.5. Compactly generated groupoids 243 x3.6. Hyperbolic groupoids 252 Exercises 252 iii iv Contents Chapter 4. Iterated monodromy groups 255 x4.1. Iterated monodromy groups of self-coverings 255 x4.2. Self-similar groups 265 x4.3. General case 274 x4.4. Expanding maps and contracting groups 291 x4.5. Thurston maps and related structures 311 x4.6. Iterations of polynomials 333 x4.7. Functoriality 334 Exercises 340 Chapter 5. Groups from groupoids 345 x5.1. Full groups 345 x5.2. AF groupoids 364 x5.3. Homology of totally disconnected ´etalegroupoids 371 x5.4. Almost finite groupoids 378 x5.5. Bounded type 380 x5.6. Torsion groups 395 x5.7. Fragmentations of dihedral groups 401 x5.8. Purely infinite groupoids 423 Exercises 424 Chapter 6. Growth and amenability 427 x6.1. Growth of groups 427 x6.2. -
Nonlocally Maximal Hyperbolic Sets for Flows
Brigham Young University BYU ScholarsArchive Theses and Dissertations 2015-06-01 Nonlocally Maximal Hyperbolic Sets for Flows Taylor Michael Petty Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mathematics Commons BYU ScholarsArchive Citation Petty, Taylor Michael, "Nonlocally Maximal Hyperbolic Sets for Flows" (2015). Theses and Dissertations. 5558. https://scholarsarchive.byu.edu/etd/5558 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. Nonlocally Maximal Hyperbolic Sets for Flows Taylor Michael Petty A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Todd Fisher, Chair Lennard F. Bakker Christopher P. Grant Department of Mathematics Brigham Young University June 2015 Copyright c 2015 Taylor Michael Petty All Rights Reserved abstract Nonlocally Maximal Hyperbolic Sets for Flows Taylor Michael Petty Department of Mathematics, BYU Master of Science In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one. -
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. -
Symbolic Dynamics and Markov Partitions
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 35, Number 1, January 1998, Pages 1–56 S 0273-0979(98)00737-X SYMBOLIC DYNAMICS AND MARKOV PARTITIONS ROY L. ADLER Abstract. The decimal expansion of real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by sym- bolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by sym- bolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely, hyperbolic automorphisms of the two dimensional torus. While there are some results in higher dimensions, this area remains a fertile one for research. 1. Introduction We address the question: how and to what extent can a dynamical system be represented by a symbolic one? The first use of infinite sequences of symbols to describe orbits is attributed to a nineteenth century work of Hadamard [H]. How- ever the present work is rooted in something very much older and familiar to us all: namely, the representation of real numbers by infinite binary expansions. As the example of Section 3.2 shows, such a representation is related to the behavior of a special partition under the action of a map. Partitions of this sort have been linked to the name Markov because of their connection to discrete time Markov processes. -
Dynamical Systems and Ergodic Theory
MATH36206 - MATHM6206 Dynamical Systems and Ergodic Theory Teaching Block 1, 2017/18 Lecturer: Prof. Alexander Gorodnik PART III: LECTURES 16{30 course web site: people.maths.bris.ac.uk/∼mazag/ds17/ Copyright c University of Bristol 2010 & 2016. This material is copyright of the University. It is provided exclusively for educational purposes at the University and is to be downloaded or copied for your private study only. Chapter 3 Ergodic Theory In this last part of our course we will introduce the main ideas and concepts in ergodic theory. Ergodic theory is a branch of dynamical systems which has strict connections with analysis and probability theory. The discrete dynamical systems f : X X studied in topological dynamics were continuous maps f on metric spaces X (or more in general, topological→ spaces). In ergodic theory, f : X X will be a measure-preserving map on a measure space X (we will see the corresponding definitions below).→ While the focus in topological dynamics was to understand the qualitative behavior (for example, periodicity or density) of all orbits, in ergodic theory we will not study all orbits, but only typical1 orbits, but will investigate more quantitative dynamical properties, as frequencies of visits, equidistribution and mixing. An example of a basic question studied in ergodic theory is the following. Let A X be a subset of O+ ⊂ the space X. Consider the visits of an orbit f (x) to the set A. If we consider a finite orbit segment x, f(x),...,f n−1(x) , the number of visits to A up to time n is given by { } Card 0 k n 1, f k(x) A .