DOCSLIB.ORG

Dynamical Systems and a Brief Introduction to Ergodic Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dynamical Systems and a Brief Introduction to Ergodic Theory Dynamical Systems and a Brief Introduction to Ergodic Theory Leo Baran Spring 2014 Abstract This paper explores dynamical systems of different types and orders, culminating in an ex- amination of the properties of the logistic map. It also introduces Ergodic theory and important results in the field. Contents 1 Dynamical Systems4 1 1.1 Differential Equations . 2 1 1.1.1 Flows on R ..................................... 3 2 1.1.2 Flows on R ..................................... 5 3 1.1.3 R and chaos . 8 1.2 Maps . 9 1.2.1 The Logistic Map . 9 2 Ergodic Theory 12 2.1 Measure Theory Preliminaries1;2 ............................. 12 2.2 A Few Results3 ....................................... 13 1 Dynamical Systems4 The field of dynamics came to be in the 1600s by Newton's developments in differential equations and their applications to the laws of gravitation and planetary motion. He realized that finding an 1 exact solution for some dynamical systems (such as the three-body problem, or describing the exact motion of three planetary bodies confined to the laws of gravitation) was essentially impossible. A coupld centuries later, Poincare developed a novel approach to analyzing such systems. His method was to answer qualitative questions about the system rather than develop an exact quantitative solution. Until the mid-1900s, dynamics was largely concerned with non-linear oscillating systems and their applications to physics. The development of high-speed computing led Lorenz in 1963 to discover chaotic motion in dynamics. Since then interest in dynamics and chaos has proliferated and applications to real-world systems have become exceedingly numerous. There are two main types of dynamical systems: differential equations and iterated maps. Dif- ferential equations describe the motion of systems in continuous time, while iterated maps deal exclusively with discrete time. 1.1 Differential Equations Consider an n-dimensional space whose points (x1(t); x2(t); : : : ; xn(t)) are functions of time. In general, an autonomous dynamical system on this n-dimensional phase space is defined as the system d x = f (x ; x ; : : : ; x ) dt 1 1 1 2 n d x = f (x ; x ; : : : ; x ) dt 2 2 1 2 n ::: d x = f (x ; x ; : : : ; x ) dt n n 1 2 n Where autonomous refers to the fact that f1; : : : ; fn don't depend on t. In this case, the differ- ential equations that compose the system are of first order, i.e. they only use first time derivatives of (x1; x2; : : : ; xn). 2 1 1.1.1 Flows on R On the other hand, in dynamical systems a first-order system is of the form d x(t) = f(x(t)) dt Example 1.1. x0(t) = sin(x(t)) In this case we can find an exact solution for the system by doing some slightly sketchy separating of variables: dx dx Z Z dx = sin(x) , dt = , dt = , t = − ln j csc(x) + cot(x)j + C: dt sin(x) sin(x) If x(0) = xo, then 0 = − ln j csc(x0) + cot(x0)j + C , C = ln j csc(x0) + cot(x0)j. Thus j csc(x ) + cot(x )j t = − ln j csc(x) + cot(x)j + ln j csc(x ) + cot(x )j = ln 0 0 0 0 j csc(x) + cot(x)j Most of the time we want to know what x(t) does as t ! 1 for an arbitrary initial condition x0, however in this case (and in most systems) determining a formula for x(t) isn't trivial. Instead we dx can interpret f(x(t)) as a vector field: t is time, x(t) is the position on R, and dt is the velocity on the line. We can graph this vector field while keeping in mind that a positive velocity means we move in the positive direction on R and vice versa for negative. For every point at which x0(t) = 0, x(t) is stationary. These points are called fixed points. In the above graph, certain fixed points are marked with solid circles and others with hollow circles. The fixed points with solid circles at (2n+1)π; n 2 Z are points toward which x(t) is attracted, while the hollow circles at 2nπ; n 2 Z are points from which x(t) is repelled. These points are conveniently called attracting and repelling fixed points, respectively. The fixed to which x(t) tends depends on 3 π −π the initial condition. If x(0) = 4 ; then the system will tend to π as t ! 1. If x(0) = 4 ; then the system will tend to −π and if x(0) = 0; the system will stay at x(t) = 0 8t: We, being mathematicians, would like to know about solutions to all dynamical systems of this form. Using the fact that our dynamical system is essentially a differential equation, we can show that solutions to nonautonomous dynamical systems exist and are unique in a certain inverval of time. Theorem 1.1. (Linear Fundamental Existence and Uniqueness Theorem) Let f and g be continuous functions on (a; b) 2 R and t0 2 (a; b) and x0 2 R. Then there is a unique function p(t) = x that satisfies 0 x = f(t)x + g(t); p(t0) = x0 on (a; b). Namely, Z t −F (t) −F (t) F (τ) p(t) = x0e + e g(τ)e dτ t0 Where F (t) = R t f(τ)dτ. t0 Proof. First we prove existence. Suppose the solution p(t) takes the form above. Take a first deriva- tive: p0(t) = −x e−F (t) d [F (t)]+ d [e−F (t)] R t g(τ)eF (τ)dτ+ d [R t g(τ)eF (τ)dτ]e−F (t) = −x e−F (t)f(t)− 0 dt dt t0 dt t0 0 f(t)e−F (t) R t g(τ)eF (τ)dτ+g(t)eF (t)e−F (t) = −f(t)[x e−F (t)+e−F (t) R t g(τ)eF (τ)dτ]+g(t) = −f(t)p(t)+ t0 0 t0 t g(t) , p0(t) = −f(t)p(t)+g(t) , p0(t)+f(t)p(t) = g(t). Also p(t ) = x e−F (t0)+e−F (t0) R 0 g(τ)eF (τ)dτ = 0 0 t0 R t0 t f(τ)dτ x0e 0 = x0. Thus p(t) is a solution to the initial value problem. Next we show that the solution is unique. Suppose q(t) is also a solution to the problem. Suppose F (t) 0 0 F (t) F (t) F (t) 0 r(t) = q(t)e . Then r(t0) = q(t0) = x0. Also r (t) = q (t)e − q(t)f(t)e = e [q (t) − q(t)f(t)] = eF (t)g(t). We know by the fundamental theorem of calculus that R t r0(τ)dτ = r(t) − t0 r(t ) , r(t) = r(t ) + R t eF (τ)g(τ)dτ = x + R t eF (τ)g(τ)dτ. 0 0 t0 0 t0 We also know that r(t) = q(t)eF (t) , q(t) = r(t)e−F (t). Thus q(t) = e−F (t)[x +R t eF (τ)g(τ)dτ] = 0 t0 x e−F (t) + e−F (t) R t eF (τ)g(τ)dτ. 0 t0 We therefore know that every solution to the problem must take this form, so the solution is unique. 4 Dynamical systems in two dimensions become a bit more interesting. 2 1.1.2 Flows on R We consider the general system d x (t) = f (x (t); x (t)) dt 1 1 1 2 d x (t) = f (x (t); x (t)) dt 2 2 1 2 Or d x = f(x) dt If our system is linear, then d d x (t) = ax (t) + bx (t); x (t) = cx (t) + dx (t) , x_ = Ax dt 1 1 2 dt 2 1 2 where A = ( a b ) and x = ( x1(t) ) c d x2(t) We can learn a great deal about the behavior of this system by studying the matrix A. We would like to find lines on the plane which are invariant under A. That is, Av = λv. (Like usual in solving differential equations, we assume that the solution takes the form x(t) = eλtv.) This means we want to find the eigenvalues λj and and eigenvectors v 6= 0 of A. We find them in the following way: Av = λv , (A − λI)v = 0 =) det A − λI = 0. The last step is derived from the fact that v is assumed not to be zero, thus (A − λI) is a non-invertible matrix, which means its a−λ b 2 determinant is zero. Next we solve det A − λI = 0 , det( c d−λ ) = 0 , λ − τλ + ∆ = 0 where τ = a + d = Trace(A) and ∆ = ad − bc = det(A). Using the quadratic equation, we see that p p τ + τ 2 − 4∆ τ − τ 2 − 4∆ λ = ; λ = 1 2 2 2 are the two eigenvalues of A corresponding to eigenvectors v1; v2: The general solution to the system λ t λ t is a linear combination of the two solutions: x(t) = c1e 1 v1 + c2e 1 v2. We can use the following result in linear algebra to see where this solution exists. 5 Lemma 1.2. Suppose A is a 2 by 2 nonzero matrix with eigenvalues λ1 and λ2 such that λ1 6= λ2. Then the eigenvectors v1 and v2 that correspond to λ1 and λ2 are linearly independent. Proof.
Load more

Recommended publications
  • Ergodic Theory
    MATH41112/61112 Ergodic Theory Charles Walkden 4th January, 2018 MATH4/61112 Contents Contents 0 Preliminaries 2 1 An introduction to ergodic theory. Uniform distribution of real se- quences 4 2 More on uniform distribution mod 1. Measure spaces. 13 3 Lebesgue integration. Invariant measures 23 4 More examples of invariant measures 38 5 Ergodic measures: definition, criteria, and basic examples 43 6 Ergodic measures: Using the Hahn-Kolmogorov Extension Theorem to prove ergodicity 53 7 Continuous transformations on compact metric spaces 62 8 Ergodic measures for continuous transformations 72 9 Recurrence 83 10 Birkhoff’s Ergodic Theorem 89 11 Applications of Birkhoff’s Ergodic Theorem 99 12 Solutions to the Exercises 108 1 MATH4/61112 0. Preliminaries 0. Preliminaries 0.1 Contact details § The lecturer is Dr Charles Walkden, Room 2.241, Tel: 0161 275 5805, Email: [email protected]. My office hour is: Monday 2pm-3pm. If you want to see me at another time then please email me first to arrange a mutually convenient time. 0.2 Course structure § This is a reading course, supported by one lecture per week. I have split the notes into weekly sections. You are expected to have read through the material before the lecture, and then go over it again afterwards in your own time. In the lectures I will highlight the most important parts, explain the statements of the theorems and what they mean in practice, and point out common misunderstandings. As a general rule, I will not normally go through the proofs in great detail (but they are examinable unless indicated otherwise).
    [Show full text]
  • MA427 Ergodic Theory
    MA427 Ergodic Theory Course Notes (2012-13) 1 Introduction 1.1 Orbits Let X be a mathematical space. For example, X could be the unit interval [0; 1], a circle, a torus, or something far more complicated like a Cantor set. Let T : X ! X be a function that maps X into itself. Let x 2 X be a point. We can repeatedly apply the map T to the point x to obtain the sequence: fx; T (x);T (T (x));T (T (T (x))); : : : ; :::g: We will often write T n(x) = T (··· (T (T (x)))) (n times). The sequence of points x; T (x);T 2(x);::: is called the orbit of x. We think of applying the map T as the passage of time. Thus we think of T (x) as where the point x has moved to after time 1, T 2(x) is where the point x has moved to after time 2, etc. Some points x 2 X return to where they started. That is, T n(x) = x for some n > 1. We say that such a point x is periodic with period n. By way of contrast, points may move move densely around the space X. (A sequence is said to be dense if (loosely speaking) it comes arbitrarily close to every point of X.) If we take two points x; y of X that start very close then their orbits will initially be close. However, it often happens that in the long term their orbits move apart and indeed become dramatically different.
    [Show full text]
  • Conformal Fractals – Ergodic Theory Methods
    Conformal Fractals – Ergodic Theory Methods Feliks Przytycki Mariusz Urba´nski May 17, 2009 2 Contents Introduction 7 0 Basic examples and definitions 15 1 Measure preserving endomorphisms 25 1.1 Measurespacesandmartingaletheorem . 25 1.2 Measure preserving endomorphisms, ergodicity . .. 28 1.3 Entropyofpartition ........................ 34 1.4 Entropyofendomorphism . 37 1.5 Shannon-Mcmillan-Breimantheorem . 41 1.6 Lebesguespaces............................ 44 1.7 Rohlinnaturalextension . 48 1.8 Generalized entropy, convergence theorems . .. 54 1.9 Countabletoonemaps.. .. .. .. .. .. .. .. .. .. 58 1.10 Mixingproperties . .. .. .. .. .. .. .. .. .. .. 61 1.11 Probabilitylawsand Bernoulliproperty . 63 Exercises .................................. 68 Bibliographicalnotes. 73 2 Compact metric spaces 75 2.1 Invariantmeasures . .. .. .. .. .. .. .. .. .. .. 75 2.2 Topological pressure and topological entropy . ... 83 2.3 Pressureoncompactmetricspaces . 87 2.4 VariationalPrinciple . 89 2.5 Equilibrium states and expansive maps . 94 2.6 Functionalanalysisapproach . 97 Exercises ..................................106 Bibliographicalnotes. 109 3 Distance expanding maps 111 3.1 Distance expanding open maps, basic properties . 112 3.2 Shadowingofpseudoorbits . 114 3.3 Spectral decomposition. Mixing properties . 116 3.4 H¨older continuous functions . 122 3 4 CONTENTS 3.5 Markov partitions and symbolic representation . 127 3.6 Expansive maps are expanding in some metric . 134 Exercises .................................. 136 Bibliographicalnotes. 138 4
    [Show full text]
  • AN INTRODUCTION to DYNAMICAL BILLIARDS Contents 1
    AN INTRODUCTION TO DYNAMICAL BILLIARDS SUN WOO PARK 2 Abstract. Some billiard tables in R contain crucial references to dynamical systems but can be analyzed with Euclidean geometry. In this expository paper, we will analyze billiard trajectories in circles, circular rings, and ellipses as well as relate their charactersitics to ergodic theory and dynamical systems. Contents 1. Background 1 1.1. Recurrence 1 1.2. Invariance and Ergodicity 2 1.3. Rotation 3 2. Dynamical Billiards 4 2.1. Circle 5 2.2. Circular Ring 7 2.3. Ellipse 9 2.4. Completely Integrable 14 Acknowledgments 15 References 15 Dynamical billiards exhibits crucial characteristics related to dynamical systems. Some billiard tables in R2 can be understood with Euclidean geometry. In this ex- pository paper, we will analyze some of the billiard tables in R2, specifically circles, circular rings, and ellipses. In the first section we will present some preliminary background. In the second section we will analyze billiard trajectories in the afore- mentioned billiard tables and relate their characteristics with dynamical systems. We will also briefly discuss the notion of completely integrable billiard mappings and Birkhoff's conjecture. 1. Background (This section follows Chapter 1 and 2 of Chernov [1] and Chapter 3 and 4 of Rudin [2]) In this section, we define basic concepts in measure theory and ergodic theory. We will focus on probability measures, related theorems, and recurrent sets on certain maps. The definitions of probability measures and σ-algebra are in Chapter 1 of Chernov [1]. 1.1. Recurrence. Definition 1.1. Let (X,A,µ) and (Y ,B,υ) be measure spaces.
    [Show full text]
  • Ergodic Theory, Fractal Tops and Colour Stealing 1
    ERGODIC THEORY, FRACTAL TOPS AND COLOUR STEALING MICHAEL BARNSLEY Abstract. A new structure that may be associated with IFS and superIFS is described. In computer graphics applications this structure can be rendered using a new algorithm, called the “colour stealing”. 1. Ergodic Theory and Fractal Tops The goal of this lecture is to describe informally some recent realizations and work in progress concerning IFS theory with application to the geometric modelling and assignment of colours to IFS fractals and superfractals. The results will be described in the simplest setting of a single IFS with probabilities, but many generalizations are possible, most notably to superfractals. Let the iterated function system (IFS) be denoted (1.1) X; f1, ..., fN ; p1,...,pN . { } This consists of a finite set of contraction mappings (1.2) fn : X X,n=1, 2, ..., N → acting on the compact metric space (1.3) (X,d) with metric d so that for some (1.4) 0 l<1 ≤ we have (1.5) d(fn(x),fn(y)) l d(x, y) ≤ · for all x, y X.Thepn’s are strictly positive probabilities with ∈ (1.6) pn =1. n X The probability pn is associated with the map fn. We begin by reviewing the two standard structures (one and two)thatare associated with the IFS 1.1, namely its set attractor and its measure attractor, with emphasis on the Collage Property, described below. This property is of particular interest for geometrical modelling and computer graphics applications because it is the key to designing IFSs with attractor structures that model given inputs.
    [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]
  • Ergodic Theory Approach to Chaos: Remarks and Computational Aspects
    Int. J. Appl. Math. Comput. Sci., 2012, Vol. 22, No. 2, 259–267 DOI: 10.2478/v10006-012-0019-4 ERGODIC THEORY APPROACH TO CHAOS: REMARKS AND COMPUTATIONAL ASPECTS ∗ ∗∗ PAWEŁ J. MITKOWSKI ,WOJCIECH MITKOWSKI ∗ Faculty of Electrical Engineering, Automatics, Computer Science and Electronics AGH University of Science and Technology, al. Mickiewicza 30/B-1, 30-059 Cracow, Poland e-mail: [email protected] ∗∗Department of Automatics AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Cracow, Poland e-mail: [email protected] We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the pre- viously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota’s conjecture concerning nontrivial ergodic properties of the model. Keywords: ergodic theory, chaos, invariant measures, attractors, delay differential equations. 1. Introduction man, 2001). Transformations (or flows) with an invariant measure display three main levels of irregular behaviour, In the literature concerning dynamical systems we can i.e., (ranging from the lowest to the highest) ergodicity, find many definitions of chaos in various approaches mixing and exactness. Between ergodicity and mixing we (Rudnicki, 2004; Devaney, 1987; Bronsztejn et al., 2004). can also distinguish light mixing, mild mixing and weak Our central issue here will be the ergodic theory appro- mixing (Lasota and Mackey, 1994; Silva, 2010) and, on ach.
    [Show full text]
  • STABLE ERGODICITY 1. Introduction a Dynamical System Is Ergodic If It
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 1, Pages 1{41 S 0273-0979(03)00998-4 Article electronically published on November 4, 2003 STABLE ERGODICITY CHARLES PUGH, MICHAEL SHUB, AND AN APPENDIX BY ALEXANDER STARKOV 1. Introduction A dynamical system is ergodic if it preserves a measure and each measurable invariant set is a zero set or the complement of a zero set. No measurable invariant set has intermediate measure. See also Section 6. The classic real world example of ergodicity is how gas particles mix. At time zero, chambers of oxygen and nitrogen are separated by a wall. When the wall is removed, the gasses mix thoroughly as time tends to infinity. In contrast think of the rotation of a sphere. All points move along latitudes, and ergodicity fails due to existence of invariant equatorial bands. Ergodicity is stable if it persists under perturbation of the dynamical system. In this paper we ask: \How common are ergodicity and stable ergodicity?" and we propose an answer along the lines of the Boltzmann hypothesis { \very." There are two competing forces that govern ergodicity { hyperbolicity and the Kolmogorov-Arnold-Moser (KAM) phenomenon. The former promotes ergodicity and the latter impedes it. One of the striking applications of KAM theory and its more recent variants is the existence of open sets of volume preserving dynamical systems, each of which possesses a positive measure set of invariant tori and hence fails to be ergodic. Stable ergodicity fails dramatically for these systems. But does the lack of ergodicity persist if the system is weakly coupled to another? That is, what happens if you have a KAM system or one of its perturbations that refuses to be ergodic, due to these positive measure sets of invariant tori, but somewhere in the universe there is a hyperbolic or partially hyperbolic system weakly coupled to it? Does the lack of egrodicity persist? The answer is \no," at least under reasonable conditions on the hyperbolic factor.
    [Show full text]
  • Ergodic Theory of Chaotic Dynamical Systems
    Ergodic Theory of Chaotic Dynamical Systems Lai-Sang Young 1 2 This is the text of the author’s plenary lecture at the International Congress of Mathematical Physics in 1997 This article is about the ergodic theory of differentiable dynamical systems in finite dimensions. Limiting our discussions to discrete time, we are concerned with iterations of maps from Rn or finite dimensional manifolds to themselves. We consider only maps that generate chaotic dynamics, and our focus is on their statistical properties. Complex geometric behavior in dynamical systems was observed by Poincar´e, but no attempt was made to study it systematically until the 1960’s when Smale proposed the idea of a hyperbolic invariant set or “horseshoe” [37]: DEFINITION 1 A diffeomorphism f : M → M is said to be uniformly hyperbolic on a compact invariant set Λ if at each x ∈ Λ the tangent space splits into Df-invariant u s subspaces Ex and Ex with the following properties: ∃C ≥ 1 and λ> 1 independent of x n 1 n u n n s such that ∀n ≥ 0, |Dfx v| ≥ C− λ |v| for v ∈ Ex and |Dfx v| ≤ Cλ− |v| for v ∈ Ex. Hyperbolic sets were used as geometric models of complex behavior because hyper- bolicity implies a sensitive dependence on initial conditions: the orbits of most pairs of nearby points diverge exponentially fast in both forward and backward times. A system is said to satisfy Axiom A if all of its essential parts are uniformly hyperbolic (see [37]). In the 10 years or so since Smale’s invention of Axiom A, an ergodic theory for these systems emerged, due largely to Sinai [35] who first developed the theory for Anosov sys- tems and to Ruelle [32] who worked in the more general Axiom A setting using Markov partitions constructed by Bowen [7].
    [Show full text]
  • D. S. Ornstein Ergodic Theory, Randomness, and Dynamical
    270 BOOK REVIEWS ORNSTEIN, D. S., Ergodic Theory, Randomness, and Dynamical Systems (Yale University Press, 1975), £2-50. This book is an account of recent major advances in ergodic theory, due mainly to the author. In ergodic theory one considers invertible measure-preserving transforma- tions of a probability measure space; a natural problem is to classify these up to iso- morphism. In general this is hopelessly difficult, but recently much progress has been made for some important classes of transformations. The major breakthrough was Ornstein's proof in 1970 that the Bernoulli shifts were classified completely by an isomorphic invariant introduced by Kolmogorov in 1958, namely the entropy. (A Bernoulli shift is the shift transformation on the set of all 2-sided sequences of elements of a probability space {X, /J.) with the product measure. If/x is purely atomic and the measures of the atoms are p\,pi,... then the entropy of the shift is — 2/?( logp(. Otherwise it is infinite.) The techniques he developed enabled Ornstein and others to show that various classes of transformation are isomorphic to Bernoulli shifts and hence subject to the above classification. These include ergodic group automorphisms of the n-torus and various transformations arising from dynamical systems. Analogous results were found for flows (one-parameter groups of transformations). The results have some relevance to the theory of stationary stochastic processes—such a process induces a transformation on the underlying probability space, which is isomorphic to a Bernoulli shift provided the future is nearly independent of the past, in a sense which can be made precise.
    [Show full text]
  • A Simple Introduction to Ergodic Theory
    A Simple Introduction to Ergodic Theory Karma Dajani and Sjoerd Dirksin December 18, 2008 2 Contents 1 Introduction and preliminaries 5 1.1 What is Ergodic Theory? . 5 1.2 Measure Preserving Transformations . 6 1.3 Basic Examples . 9 1.4 Recurrence . 15 1.5 Induced and Integral Transformations . 15 1.5.1 Induced Transformations . 15 1.5.2 Integral Transformations . 18 1.6 Ergodicity . 20 1.7 Other Characterizations of Ergodicity . 22 1.8 Examples of Ergodic Transformations . 24 2 The Ergodic Theorem 29 2.1 The Ergodic Theorem and its consequences . 29 2.2 Characterization of Irreducible Markov Chains . 41 2.3 Mixing . 45 3 Measure Preserving Isomorphisms and Factor Maps 47 3.1 Measure Preserving Isomorphisms . 47 3.2 Factor Maps . 51 3.3 Natural Extensions . 52 4 Entropy 55 4.1 Randomness and Information . 55 4.2 Definitions and Properties . 56 4.3 Calculation of Entropy and Examples . 63 4.4 The Shannon-McMillan-Breiman Theorem . 66 4.5 Lochs’ Theorem . 72 3 4 5 Hurewicz Ergodic Theorem 79 5.1 Equivalent measures . 79 5.2 Non-singular and conservative transformations . 80 5.3 Hurewicz Ergodic Theorem . 83 6 Invariant Measures for Continuous Transformations 89 6.1 Existence . 89 6.2 Unique Ergodicity . 96 7 Topological Dynamics 101 7.1 Basic Notions . 102 7.2 Topological Entropy . 108 7.2.1 Two Definitions . 108 7.2.2 Equivalence of the two Definitions . 114 7.3 Examples . 118 8 The Variational Principle 127 8.1 Main Theorem . 127 8.2 Measures of Maximal Entropy . 136 Chapter 1 Introduction and preliminaries 1.1 What is Ergodic Theory? It is not easy to give a simple definition of Ergodic Theory because it uses techniques and examples from many fields such as probability theory, statis- tical mechanics, number theory, vector fields on manifolds, group actions of homogeneous spaces and many more.
    [Show full text]
  • ERGODIC and SPECTRAL PROPERTIES an Interval Exchange
    STRUCTURE OF THREE-INTERVAL EXCHANGE TRANSFORMATIONS III: ERGODIC AND SPECTRAL PROPERTIES SEBASTIEN´ FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI ABSTRACT. In this paper we present a detailed study of the spectral/ergodic properties of three- interval exchange transformations. Our approach is mostly combinatorial, and relies on the dio- phantine results in Part I and the combinatorial description in Part II. We define a recursive method of generating three sequences of nested Rokhlin stacks which describe the system from a measure- theoretic point of view and which in turn gives an explicit characterization of the eigenvalues. We obtain necessary and sufficient conditions for weak mixing which, in addition to unifying all previ- ously known examples, allow us to exhibit new examples of weakly mixing three-interval exchanges. Finally we give affirmative answers to two questions posed by W.A. Veech on the existence of three- interval exchanges having irrational eigenvalues and discrete spectrum. 1. INTRODUCTION An interval exchange transformation is given by a probability vector (α1, α2, . , αk) together with a permutation π of {1, 2, . , k}. The unit interval [0, 1) is partitioned into k sub-intervals of lengths α1, α2, . , αk which are then rearranged according to the permutation π. Katok and Stepin [20] used interval exchanges to exhibit a class of systems having simple continuous spectrum. Interval exchange transformations have been extensively studied by several people including Keane [21] [22], Keynes and Newton [23], Veech [31] to [36], Rauzy [28], Masur [24], Del Junco [14], Boshernitzan [5, 6], Nogueira and Rudolph [27], Boshernitzan and Carroll [7], Berthe,´ Chekhova, and Ferenczi [4], Chaves and Nogueira [10] and Boshernitzan and Nogueira [8].
    [Show full text]