Ergodic theory concerns with the study of the long-time behavior of a dynamical system. An interesting result known as Birkhoff’s ergodic theorem states that under certain conditions, the time average exists and is equal to the space average. The applications of ergodic theory are the main concern of this note. We will introduce fundamental concepts in ergodic theory, Birkhoff’s ergodic theorem and its consequences.

An Introduction to Ergodic Theory

I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.

Random Dynamical Systems

This book is an updated version of the information theory classic, first published in 1990. About one-third of the book is devoted to Shannon source and channel coding theorems; the remainder addresses sources, channels, and codes and on information and distortion measures and their properties. New in this edition:Expanded treatment of stationary or sliding-block codes and their relations to traditional block codesExpanded discussion of results from ergodic theory relevant to information theoryExpanded treatment of B-processes -- processes formed by stationary coding memoryless sourcesNew material on trading off information and distortion, including the Marton inequalityNew material on the properties of optimal and asymptotically optimal source codesNew material on the relationships of source coding and rate-constrained simulation or modeling of random processesSignificant material not covered in other information theory texts includes stationary/sliding-block codes, a geometric view of information theory provided by process distance measures, and general Shannon coding theorems for asymptotic mean stationary sources, which may be neither ergodic nor stationary, and d-bar continuous channels.

Entropy and information theory

Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory

The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function’s structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and by constructing a stochastic differential equation model. We derive a notion of the coherence of a signal with respect to a dictionary from our characterization of the approximation errors of a pursuit. The dictionary elements slected during the initial iterations of a pursuit correspond to a function’s coherent structures. The tail of the expansion, on the other hand, corresponds to a noise which is characterized by the invariant measure of the pursuit map. When using a suitable dictionary, the expansion of a function into its coherent structures yields a compact approximation. We demonstrate a denoising algorithm based on coherent function expansions.

Adaptive greedy approximations

This text provides an introduction to ergodic theory suitable for readers knowing basic measure theory. The mathematical prerequisites are summarized in Chapter 0. It is hoped the reader will be ready to tackle research papers after reading the book. The first part of the text is concerned with measure-preserving transformations of probability spaces; recurrence properties, mixing properties, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy theory are discussed. Some examples are described and are studied in detail when new properties are presented. The second part of the text focuses on the ergodic theory of continuous transformations of compact metrizable spaces. The family of invariant probability measures for such a transformation is studied and related to properties of the transformation such as topological traitivity, minimality, the size of the non-wandering set, and existence of periodic points. Topological entropy is introduced and related to measure-theoretic entropy. Topological pressure and equilibrium states are discussed, and a proof is given of the variational principle that relates pressure to measure-theoretic entropies. Several examples are studied in detail. The final chapter outlines significant results and some applications of ergodic theory to other branches of mathematics.

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On the pseudo orbit tracing property and its relationship to stability

Preliminaries.- Measure-preserving transformations.- Isomorphism and spectral invariants.- Measure-preserving transformations with pure point spectrum.- Entropy.- Topological dynamics.- Topological entropy.

Ergodic theory: Introductory lectures

0. Introduction. Considerable progress has been made in the classification of measure preserving transformations during the last thirteen years, reaching a high point with the recent work of Ornstein [1]. Most of this theory has concentrated on invertible transformations (automorphisms) since it was here that the essential problems awaited solution. Viewed as two-sided shifts on symbol spaces, an isomorphism between invertible transformations amounts to a faithful coding between their respective infinite messages. A new problem appears, however, if a coding which does not anticipate the future is required. From this point of view such a coding establishes a correspondence between their associated one-sided shifts. This is one motivation for pursuing the classification of transformations which are not necessarily invertible (endomorphisms). No proofs will be given in this paper although it should be noted that one of the principal invariants mentioned here appears in [2]. We should also like to refer interested readers to the recent work of Versik [3], [4], to Rohlin's paper [5] and the closing remarks of Rohlin in [6]. (The question raised in the last paragraph of [6] has a negative answer.)

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Endomorphisms of a Lebesgue space

We investigate the question: if T:X → X is a uniquely ergodic homeomorphism of a compact metrizable space and B:X → GL(k,R) is a continuous map of X into the space of invertible, k × k, real matrices does \(\frac{1}{n}\log (||\mathop \Pi \limits_{i = 0}^{n - 1} B (T^i x)||)\) converge uniformly to a constant? Conditions on B are given so that the answer is ‘yes’, and an example is given to show the general answer is ‘no’ when k≥2. The more general case of vector bundle automorphisms covering T is considered.

Peter Walters

Papers

An Introduction to Ergodic Theory

On the pseudo orbit tracing property and its relationship to stability

Ergodic theory: Introductory lectures

Endomorphisms of a Lebesgue space

Unique ergodicity and random matrix products