An Introduction to Ergodic Theory

Author: Peter Walters
Publisher: Springer Science & Business Media
ISBN: 9780387951522
Format: PDF, Docs
Download Now
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.

Ergodic Theory

Author: Manfred Einsiedler
Publisher: Springer Science & Business Media
ISBN: 9780857290212
Format: PDF, Kindle
Download Now
This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits. Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

Invitation to Ergodic Theory

Author: César Ernesto Silva
Publisher: American Mathematical Soc.
ISBN: 0821844202
Format: PDF, Kindle
Download Now
This book is an introduction to basic concepts in ergodic theory such as recurrence, ergodicity, the ergodic theorem, mixing, and weak mixing. It does not assume knowledge of measure theory; all the results needed from measure theory are presented from scratch. In particular, the book includes a detailed construction of the Lebesgue measure on the real line and an introduction to measure spaces up to the Caratheodory extension theorem. It also develops the Lebesgue theory of integration, including the dominated convergence theorem and an introduction to the Lebesgue $L^p$spaces. Several examples of a dynamical system are developed in detail to illustrate various dynamical concepts. These include in particular the baker's transformation, irrational rotations, the dyadic odometer, the Hajian-Kakutani transformation, the Gauss transformation, and the Chacon transformation. There is a detailed discussion of cutting and stacking transformations in ergodic theory. The book includes several exercises and some open questions to give the flavor of current research. The book also introduces some notions from topological dynamics, such as minimality, transitivity and symbolic spaces; and develops some metric topology, including the Baire category theorem.

Operator Theoretic Aspects of Ergodic Theory

Author: Tanja Eisner
Publisher: Springer
ISBN: 3319168983
Format: PDF, ePub, Docs
Download Now
Stunning recent results by Host–Kra, Green–Tao, and others, highlight the timeliness of this systematic introduction to classical ergodic theory using the tools of operator theory. Assuming no prior exposure to ergodic theory, this book provides a modern foundation for introductory courses on ergodic theory, especially for students or researchers with an interest in functional analysis. While basic analytic notions and results are reviewed in several appendices, more advanced operator theoretic topics are developed in detail, even beyond their immediate connection with ergodic theory. As a consequence, the book is also suitable for advanced or special-topic courses on functional analysis with applications to ergodic theory. Topics include: • an intuitive introduction to ergodic theory • an introduction to the basic notions, constructions, and standard examples of topological dynamical systems • Koopman operators, Banach lattices, lattice and algebra homomorphisms, and the Gelfand–Naimark theorem • measure-preserving dynamical systems • von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theorem • strongly and weakly mixing systems • an examination of notions of isomorphism for measure-preserving systems • Markov operators, and the related concept of a factor of a measure preserving system • compact groups and semigroups, and a powerful tool in their study, the Jacobs–de Leeuw–Glicksberg decomposition • an introduction to the spectral theory of dynamical systems, the theorems of Furstenberg and Weiss on multiple recurrence, and applications of dynamical systems to combinatorics (theorems of van der Waerden, Gallai,and Hindman, Furstenberg’s Correspondence Principle, theorems of Roth and Furstenberg–Sárközy) Beyond its use in the classroom, Operator Theoretic Aspects of Ergodic Theory can serve as a valuable foundation for doing research at the intersection of ergodic theory and operator theory

Foundations of Ergodic Theory

Author: Marcelo Viana
Publisher: Cambridge University Press
ISBN: 1107126967
Format: PDF, Docs
Download Now
Self-contained introductory textbook suitable for a variety of one- or two-semester courses. Rich with examples, applications and exercises.

Equilibrium States in Ergodic Theory

Author: Gerhard Keller
Publisher: Cambridge University Press
ISBN: 9780521595346
Format: PDF, Docs
Download Now
This book provides a detailed introduction to the ergodic theory of equilibrium states giving equal weight to two of its most important applications, namely to equilibrium statistical mechanics on lattices and to (time discrete) dynamical systems. It starts with a chapter on equilibrium states on finite probability spaces which introduces the main examples for the theory on an elementary level. After two chapters on abstract ergodic theory and entropy, equilibrium states and variational principles on compact metric spaces are introduced emphasizing their convex geometric interpretation. Stationary Gibbs measures, large deviations, the Ising model with external field, Markov measures, Sinai-Bowen-Ruelle measures for interval maps and dimension maximal measures for iterated function systems are the topics to which the general theory is applied in the last part of the book. The text is self contained except for some measure theoretic prerequisites which are listed (with references to the literature) in an appendix.

Ergodic Theory

Author: David Kerr
Publisher: Springer
ISBN: 3319498479
Format: PDF, Docs
Download Now
This book provides an introduction to the ergodic theory and topological dynamics of actions of countable groups. It is organized around the theme of probabilistic and combinatorial independence, and highlights the complementary roles of the asymptotic and the perturbative in its comprehensive treatment of the core concepts of weak mixing, compactness, entropy, and amenability. The more advanced material includes Popa's cocycle superrigidity, the Furstenberg-Zimmer structure theorem, and sofic entropy. The structure of the book is designed to be flexible enough to serve a variety of readers. The discussion of dynamics is developed from scratch assuming some rudimentary functional analysis, measure theory, and topology, and parts of the text can be used as an introductory course. Researchers in ergodic theory and related areas will also find the book valuable as a reference.

Ergodic Theory

Author: Karl E. Petersen
Publisher: Cambridge University Press
ISBN: 9780521389976
Format: PDF, ePub
Download Now
The author presents the fundamentals of the ergodic theory of point transformations and several advanced topics of intense research. The study of dynamical systems forms a vast and rapidly developing field even when considering only activity whose methods derive mainly from measure theory and functional analysis. Each of the basic aspects of ergodic theory--examples, convergence theorems, recurrence properties, and entropy--receives a basic and a specialized treatment. The author's accessible style and the profusion of exercises, references, summaries, and historical remarks make this a useful book for graduate students or self study.

Ergodic Theory of Numbers

Author: Karma Dajani
Publisher: Cambridge University Press
ISBN: 9780883850343
Format: PDF, Mobi
Download Now
Introduction to ergodic theory of numbers for graduate students and researchers.

Introduction to Smooth Ergodic Theory

Author: Luis Barreira
Publisher: American Mathematical Soc.
ISBN: 0821898531
Format: PDF, Kindle
Download Now
This book is the first comprehensive introduction to smooth ergodic theory. It consists of two parts: the first introduces the core of the theory and the second discusses more advanced topics. In particular, the book describes the general theory of Lyapun