DOCSLIB.ORG

Topological Conjugacy for Lipschitz Perturbations of Non-Autonomous Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topological Conjugacy for Lipschitz Perturbations of Non-Autonomous Systems DISCRETE AND CONTINUOUS doi:10.3934/dcds.2016017 DYNAMICAL SYSTEMS Volume 36, Number 9, September 2016 pp. 5011{5024 TOPOLOGICAL CONJUGACY FOR LIPSCHITZ PERTURBATIONS OF NON-AUTONOMOUS SYSTEMS Ming-Chia Li∗ and Ming-Jiea Lyu Department of Applied Mathematics National Chiao Tung University, 1001 University Road Hsinchu 300, Taiwan (Communicated by Lan Wen) Abstract. In this paper, topological conjugacy for two-sided non-hyperbolic and non-autonomous discrete dynamical systems is studied. It is shown that if the system has covering relations with weak Lyapunov condition determined by a transition matrix, there exists a sequence of compact invariant sets restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by the transition matrix. Moreover, if the systems have covering relations with exponential dichotomy and small Lipschitz perturbations, then there is a constructive verification proof of the weak Lyapunov condition, and so topological dynamics of these systems are fully understood by symbolic representations. In addition, the tolerance of Lipschitz perturbation can be characterised by the dichotomy tuple . Here, the weak Lyapunov condition is adapted from [12, 24, 15] and the exponential dichotomy is from [2]. 1. Introduction. In the study of dynamical systems, getting topological conju- gacy to a symbolic system is a popular method to realize the long-term qualitative behavior of a system from topological viewpoints. Under uniformly hyperbolic con- dition, the classical results are presented by Smale [21] for horseshoe maps, Hirsch and Pugh [8] for hyperbolic maps, and Robinson [19] for structurally stable maps, etc. Without hyperbolicity, there are computable methods to obtain topological semi-conjugacy; for instance, Zgliczy´nski[22, 23] introduced covering relations, Mis- chaikov et al. [4,5, 13] used discrete Conley index, Yorke et al. [10,9] presented expanders family, and Richeson et al. [7, 17] for index systems. Furthermore to obtain topological conjugacy, in our previous work [15, 16], we brought in the Lya- punov condition, which is adapted from the cone condition in [12, 24]. In this paper, we weaken the Lyapunov condition and establish a method to verify it, but still secure the truth of the conjugacy result. Non-autonomous discrete systems are time-dependent so that the maps describ- ing the evolution are allowed to change with time; while autonomous systems are time-independent and can be viewed as a particular case of non-autonomous ones. 2010 Mathematics Subject Classification. 37B55, 37D25, 37C15, 37B10. Key words and phrases. Topological conjugacy, exponential dichotomy, Lipschitz perturbation, non-autonomous systems, nonuniformly hyperbolic systems. This work was partially supported by MOST grant 104-2115-M-009-003-MY2. The authors are grateful to the anonymous referee for valuable suggestions. ∗ Corresponding author: M.-C. Li. 5011 5012 MING-CHIA LI AND MING-JIEA LYU Such systems arise quite naturally in practical applications, for example, the Eu- ler method with a variable time-step hn > 0 for a differential equationx _ = f(x) gives a non-autonomous system of the form xn+1 = xn + hnf(xn): In recent years there has been growing interest in non-autonomous problems; refer to [1,6] for surveys with open problems and [11, 18] for monographs in this topics. For non- uniformly hyperbolic dynamics, Barreira and Valls [2] established a version of the Grobman-Hartman theorem by constructing topological conjugacy between a lin- ear non-autonomous system with non-uniform exponential dichotomy and its small Lipschitz perturbations. Moreover, in [3], they discussed the relation between non- uniform exponential dichotomy and strict Lyapunov sequence. For motivational purpose, let us illustrate interesting dynamics with the following simple example. For each integer n, let Dn be a constant such that 1 ≤ Dn ≤ 3 if n ≥ 0 and 2 ≤ Dn ≤ 3 if n < 0; and define a map fn on the real line by ( Dnx − 1 + Ln; for x 2 R and Dn ≥ 2; fn(x) = 2 2 (1) ±Dnx ± Ln x + sin (2πx) ; for x Q 0 and Dn < 2; where Ln > 0 is a small constant. Then (fn)n2Z forms a two-sided non-autonomous system with linear part (Dn)n2Z and Lipschitz perturbation involving (Ln)n2Z.A bounded orbit of the system is a two-sided bounded sequence (xn)n2Z of numbers satisfying xn+1 = fn(xn) for all integers n. It is clear that if Dn ≥ 2 for all n, then P1 Qi −1 xn ≡ −1 + i=n(Di − Li) j=n Dj is the unique bounded orbit of system (1) and it persists under small Lipschitz perturbations. If Dn = 1 for all n ≥ 0, then Pn Qi −1 xn ≡ i=−1(1 − Li) j=n Dj for n < 0 and xn ≡ 0 for n ≥ 0 form a bounded orbit for the system; nevertheless, the uniqueness of bounded orbits requires the assumption that the one-sided sequence (Ln)n≥0 is not summable, and hence it dose not necessarily persist under perturbations. For topological conjugacy from system (1) to its small perturbations, it gives rise to the question whether a unique bounded orbit persists under perturbations. It is not difficult to see that the answer is yes if the limit inferior of Dn is not equal to one as n goes to infinity, while the system is uniformly hyperbolic in some sense. However, the answer is no if the Q1 limit of Dn is equal to one as n goes to infinity, the infinite product n=0 Dn of the one-sided sequence (Dn)n≥0 is convergent, and the one-sided sequence (Ln)n≥0 is summable, while the system is neither uniformly hyperbolic nor exponentially dichotomic. This leads us to consider the general case when the system is not uniformly hyperbolic but is exponentially dichotomic. Notice that if one of Dn's is less than 2, then we cannot apply the result of [15, 16], since a Lyapunov condition of quadratic form is not satisfied. In this paper, instead of just a unique bounded orbit in the above example, we consider the general case, namely, bounded invariants sets for high-dimensional systems, possibly with both stable and unstable directions. We first use covering relations of Zgliczy´nski[22, 23] to change local coordinate along stable and unstable directions, then impose weak Lyapunov condition in form of a sequence of real- valued functions to track displacements of two orbits, and finally obtain topological conjugacy. More precisely, for a two-sided non-autonomous system having covering relations with weak Lyapunov condition determined by a transition matrix A, we show that the system has a sequence of compact invariant sets restricted to which the system is topologically-conjugate to σA, the two-sided subshift of finite type induced by A. TOPOLOGICAL CONJUGACY FOR NON-AUTONOMOUS SYSTEMS 5013 In addition, we show that if the system has linear part admitting exponential dichotomy of Barreira and Valls [2] and non-linear part with small Lipschitz pertur- bation, then the covering relations must admit the weak Lyapunov condition and hence a subsystem has dynamics exactly as subshift dynamics. Moreover, we give the tolerance of Lipschitz perturbations controlled by the dichotomy tuple. 2. Statements of definitions and main results. First, we introduce some basic notations and definitions. For a positive integer K, let RK denote the space of all K-tuples of real numbers, |·| be the Euclidean norm on RK , and let k · k denote the operator-norm on the space of linear transformations on RK induced by |·|. For x 2 RK and r > 0, we denote BK (x; r) = fz 2 RK : jz − xj < rg; for the particular case when x = 0 and r = 1, we write BK = BK (0; 1), that is, the open unit ball in RK . Moreover, for a subset M of RK , let M, int(M) and @M denote the closure, interior and boundary of S, respectively. A function from a set into itself is also called a map on the set. Let N denote the set of all non-negative integers, while Z denotes the set of all integers. Let R+ denote the non-negative real numbers. We briefly recall some definitions from [25] concerning covering relations. Definition 2.1 (See [25, Definition 6]). An h-set in RK is a quadruple consisting of the following data: • a non-empty compact subset M of RK ; • a pair of numbers u(M); s(M) 2 f0; 1; :::; kg with u(M) + s(M) = K; and K K u(M) s(M) u(M) • a homeomorphism cM : R ! R = R × R with cM (M) = B × Bs(M): For simplicity, we will denote such an h-set by M, and say that the h-set M has unstable (respectively, stable) direction, if u 6= 0 (respectively, s 6= 0); furthermore, we use the following notations: M~ = Bu(M) × Bs(M); M~ − = @Bu(M) × Bs(M); M~ + = Bu(M) × @Bs(M); − −1 ~ − + −1 ~ + the unstable boundary M = cM (M ); and the stable boundary M = cM (M ); we also writev ~ = cM (v) for v 2 M. A covering relation between two h-sets is defined as follows. Definition 2.2 (See[25, Definition 7]). Let M; N be h-sets in RK with u(M) = u(N) = u and s(M) = s(N) = s; f : M ! Ru × Rs be a continuous function, and ~ −1 ~ u s f := cN ◦ f ◦ cM : M ! R × R . We say M f-covers N, denoted by f M =) N; if the following conditions are satisfied: 1. there exists a homotopy h : [0; 1]×M~ ! Ru ×Rs such that h([0; 1]; M~ −)\N~ = ;, h([0; 1]; M~ ) \ N~ + = ;, and h(0; x~) = f~(~x) for allx ~ 2 M~ ; 2. there exists a function : Ru ! Ru such that (@Bu) ⊂ RunBu and h(1; p; q) = ( (p); 0) for all p 2 Bu and q 2 Bs; and 3.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • Topological Dynamics: a Survey
    Topological Dynamics: A Survey Ethan Akin Mathematics Department The City College 137 Street and Convent Avenue New York City, NY 10031, USA September, 2007 Topological Dynamics: Outline of Article Glossary I Introduction and History II Dynamic Relations, Invariant Sets and Lyapunov Functions III Attractors and Chain Recurrence IV Chaos and Equicontinuity V Minimality and Multiple Recurrence VI Additional Reading VII Bibliography 1 Glossary Attractor An invariant subset for a dynamical system such that points su±ciently close to the set remain close and approach the set in the limit as time tends to in¯nity. Dynamical System A model for the motion of a system through time. The time variable is either discrete, varying over the integers, or continuous, taking real values. Our systems are deterministic, rather than stochastic, so the the future states of the system are functions of the past. Equilibrium An equilibrium, or a ¯xed point, is a point which remains at rest for all time. Invariant Set A subset is invariant if the orbit of each point of the set remains in the set at all times, both positive and negative. The set is + invariant, or forward invariant, if the forward orbit of each such point remains in the set. Lyapunov Function A continuous, real-valued function on the state space is a Lyapunov function when it is non-decreasing on each orbit as time moves forward. Orbit The orbit of an initial position is the set of points through which the system moves as time varies positively and negatively through all values. The forward orbit is the subset associated with positive times.
    [Show full text]
  • THE DYNAMICAL SYSTEMS APPROACH to DIFFERENTIAL EQUATIONS INTRODUCTION the Mathematical Subject We Call Dynamical Systems Was
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 1, July 1984 THE DYNAMICAL SYSTEMS APPROACH TO DIFFERENTIAL EQUATIONS BY MORRIS W. HIRSCH1 This harmony that human intelligence believes it discovers in nature —does it exist apart from that intelligence? No, without doubt, a reality completely independent of the spirit which conceives it, sees it or feels it, is an impossibility. A world so exterior as that, even if it existed, would be forever inaccessible to us. But what we call objective reality is, in the last analysis, that which is common to several thinking beings, and could be common to all; this common part, we will see, can be nothing but the harmony expressed by mathematical laws. H. Poincaré, La valeur de la science, p. 9 ... ignorance of the roots of the subject has its price—no one denies that modern formulations are clear, elegant and precise; it's just that it's impossible to comprehend how any one ever thought of them. M. Spivak, A comprehensive introduction to differential geometry INTRODUCTION The mathematical subject we call dynamical systems was fathered by Poin­ caré, developed sturdily under Birkhoff, and has enjoyed a vigorous new growth for the last twenty years. As I try to show in Chapter I, it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself: the universe is a sytem which changes in time. Every mathematician has a particular way of thinking about mathematics, but rarely makes it explicit.
    [Show full text]
  • Arxiv:Math/0408328V1 [Math.DS] 24 Aug 2004 At2 Etn Grounds Meeting 2
    ON THE INTERPLAY BETWEEN MEASURABLE AND TOPOLOGICAL DYNAMICS E. GLASNER AND B. WEISS Contents Introduction 2 Part 1. Analogies 3 1. Poincar´erecurrence vs. Birkhoff’s recurrence 3 1.1. Poincar´erecurrence theorem and topological recurrence 3 1.2. The existence of Borel cross-sections 4 1.3. Recurrence sequences and Poincar´esequences 5 2. The equivalence of weak mixing and continuous spectrum 7 3. Disjointness: measure vs. topological 10 4. Mild mixing: measure vs. topological 12 5. Distal systems: topological vs. measure 19 6. Furstenberg-Zimmer structure theorem vs. its topological PI version 21 7. Entropy: measure and topological 22 7.1. The classical variational principle 22 7.2. Entropy pairs and UPE systems 22 7.3. A measure attaining the topological entropy of an open cover 23 7.4. The variational principle for open covers 28 7.5. Further results connecting topological and measure entropy 30 7.6. Topological determinism and zero entropy 31 Part 2. Meeting grounds 33 8. Unique ergodicity 33 arXiv:math/0408328v1 [math.DS] 24 Aug 2004 9. The relative Jewett-Krieger theorem 34 10. Models for other commutative diagrams 39 11. The Furstenberg-Weiss almost 1-1 extension theorem 40 12. Cantor minimal representations 40 13. Other related theorems 41 References 44 Date: December 7, 2003. 1 2 E. GLASNER AND B. WEISS Introduction Recurrent - wandering, conservative - dissipative, contracting - expanding, deter- ministic - chaotic, isometric - mixing, periodic - turbulent, distal - proximal, the list can go on and on. These (pairs of) words — all of which can be found in the dictio- nary — convey dynamical images and were therefore adopted by mathematicians to denote one or another mathematical aspect of a dynamical system.
    [Show full text]
  • Topological Dynamics and Group Theory
    TRANSACTIONSOF THE AMERICANMATHEMATICAL SOCIETY Volume 187, Issue 1, 1974 TOPOLOGICALDYNAMICS AND GROUP THEORY BY SHMUEL GLASNER ABSTRACT. We prove, using notions and techniques of topological dynamics, that a nonamenable group contains a finitely-generated subgroup of exponential growth. We also show that a group which belongs to a certain class, defined by means of topological dynamical properties, always contains a free subgroup on two generators. Topological dynamics can be viewed as a theory of representations of groups as homeomorphism groups. In this paper we want to emphasize this point of view and to show that information about the structure of a group can be obtained by studying such representations. One illustration of this method is the characteri- zation obtained in [7] of amenable groups as groups which admit no nontrivial strongly proximal flow. (See definitions below.) Furthermore this method yields results which do not explicitly involve notions from topological dynamics, for ex- ample using the above characterization of amenable groups we prove the theorem that a nonamenable group has a finitely generated subgroup of exponential growth. (This last result also follows from the main theorem in [8, p. 150] and Lemma 5 in [9].) The paper is arranged as follows. In § 1 we define the necessary notions from topological dynamics. In particular strongly proximal, totally proximal and extreme- ly proximal flows are defined. In the second section we prove, using the notion of a strongly proximal flow, that a nonamenable group contains a finitely generated subgroup of exponential growth [9]. In the third section we show that an extremely proximal flow is both strongly proximal and totally proximal.
    [Show full text]
  • Recent Developments in Dynamical Systems: Three Perspectives
    RECENT DEVELOPMENTS IN DYNAMICAL SYSTEMS: THREE PERSPECTIVES F. BALIBREA Universidad de Murcia, Dpto. de Matem´aticas, 30100-Murcia, Spain E-mail: [email protected] T. CARABALLO Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Apdo. de Correos 1160, 41080-Sevilla (Spain) E-mail: [email protected] P.E. KLOEDEN Institut f¨ur Mathematik, Goethe-Universit¨at, D-60054 Frankfurt am Main, Germany E-mail: [email protected] J. VALERO Univ. Miguel Hern´andez de Elche, Centro de Investigaci´on Operativa, Avda. Universidad s/n, 03202 Elche (Alicante), Spain E-mail: [email protected] Received September 15, 2009; Revised October 30, 2009 Dedicated to the memory of Valery S. Melnik The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems Keywords: Topological entropy, Li-Yorke chaos, Lyapunov exponent, continua, non- autonomous systems, difference equations, ordinary and partial differential equations, set-valued dynamical system, global attractor, non-autonomous and random pullback attractors. Mathematics Subject Classifications (2000): 35B40, 35B41, 35K40, 35K55, 37B25, 37B30, 37B40, 37B45, 37B55, 37E05, 37E15, 39A20 58C06, 60H15. 1. Introduction nected with celestial mechanics. In particular, the theory of discrete dynamical systems, which mainly uses iteration theory, is one of the most relevant The theory of dynamical systems has a long prehis- topics in the subject. tory, but essentially started in its present form with the work of Poincar´eand Birkhoff on problems con- The purpose of this tutorial is to give a par- 1 2 Balibrea, Caraballo, Kloeden & Valero tial account of the progress obtained in discrete and centers and depth of centers [Ye, 1993, Kato, 1995, continuous systems is recent years, to present some Kato, 1998, Efremova & Makhrova, 2003] on open and new problems.
    [Show full text]
  • S1. Algebraic Entropy and Topological Entropy
    S1. Algebraic Entropy and Topological Entropy Organizers: • Luigi Salce (Universit´adi Padova, Italy) • Manuel Sanchis L´opez (Universitat Jaume I de Castell´o,Spain) Speakers: 1. Llu´ısAlsed`a(Universitat Aut`onomade Barcelona, Spain) Volume entropy for minimal presentations of surface groups 2. Francisco Balibrea (Universidad de Murcia, Spain) Topological entropy and related notions 3. Federico Berlai (Universit¨atWien, Austria) Scale function and topological entropy 4. Jose S. C´anovas (Universidad Polit´ecnicade Cartagena, Spain) On entropy of fuzzy extensions of continuous maps 5. Dikran Dikranjan (Universit`adi Udine, Italy) Bridge Theorems 6. Anna Giordano Bruno (Universit`adi Udine, Italy) Algebraic entropy vs topological entropy 7. V´ıctorJim´enezL´opez (Universidad de Murcia, Spain) Can negative Schwarzian derivative be used to extract order from chaos? 8. Luigi Salce (Universit`adi Padova, Italy) The hierarchy of algebraic entropies 9. Manuel Sanchis (Universitat Jaume I de Castell´o,Spain) A notion of entropy in the realm of fuzzy metric spaces 10. Peter V´amos(University of Exeter, United Kingdom) Polyentropy, Hilbert functions and multiplicity 11. Simone Virili (Universitat Aut`onomade Barcelona, Spain) Algebraic entropy of amenable group actions 1 Volume entropy for minimal presentations of surface groups Llu´ısAlsed`a∗, David Juher, J´er^ome Los and Francesc Ma~nosas Departament de Matem`atiques,Edifici Cc, Universitat Aut`onomade Barcelona, 08913 Cerdanyola del Vall`es,Barcelona, Spain [email protected] 2010 Mathematics Subject Classification. Primary: 57M07, 57M05. Secondary: 37E10, 37B40, 37B10 We study the volume entropy of certain presentations of surface groups (which include the classical ones) introduced by J. Los [2], called minimal geometric.
    [Show full text]
  • Contributions to Ergodic Theory and Topological Dynamics : Cube Structures and Automorphisms Sebastian Andres Donoso
    Contributions to ergodic theory and topological dynamics : cube structures and automorphisms Sebastian Andres Donoso To cite this version: Sebastian Andres Donoso. Contributions to ergodic theory and topological dynamics : cube structures and automorphisms. General Mathematics [math.GM]. Université Paris-Est; Universidad de Chile, 2015. English. NNT : 2015PESC1007. tel-01271433 HAL Id: tel-01271433 https://tel.archives-ouvertes.fr/tel-01271433 Submitted on 9 Feb 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thesis in co-direction, leading to the degree of Docteur de l’Université Doctor en Ciencias de la Paris-Est, Especialité: Ingeniería, Mención Mathematiques Modelación Matemática École doctorale MSTIC Universidad de Chile Sebastián Donoso Fuentes Contributions to ergodic theory and topological dynamics: cube structures and automorphisms Jury: Romain Dujardin Examiner Université Paris-Est Marne-la-Vallée Bernard Host Advisor Université Paris-Est Marne-la-Vallée Raphaël Krikorian Examiner Université Paris VI Pierre et Marie Curie Alejandro Maass Advisor Universidad de Chile Servet Martínez Examiner Universidad de Chile Mathieu Meyer Examiner Université Paris-Est Marne-la-Vallée Jean-Paul Thouvenot Rapporteur Université Paris VI Pierre et Marie Curie Xiangdong Ye Rapporteur University of Science and Technology of China May 28, 2015 Acknowledgements First, I would to thank my advisors Bernard Host and Alejandro Maass who accepted to guide me.
    [Show full text]