Ergodic theory a probabilistic approach to dynamical systems Alex Blumenthal, Lai-Sang Young

Format:

Book

Author/Creator:

Blumenthal, Alex, author.

Young, L.-S. (Lai-Sang), author.

Series:

Texts in applied mathematics ; 2196-9949 84

Texts in Applied Mathematics 2196-9949 volume 84

Language:

English

Subjects (All):

Ergodic theory.

Physical Description:

1 online resource

Place of Publication:

Cham Springer 2026

Summary:

Ergodic theory provides a powerful lens for understanding dynamical systems, recasting disordered and seemingly random behavior in the language of probability theory. This book offers a concise, rigorous introduction to the subject, suitable both as a graduate-level textbook and as a reference for both pure and applied mathematicians. Part I (Chapters 1-7) lays the foundation, covering invariant measures, measure-theoretic isomorphisms, ergodicity, mixing, entropy, and culminating in the Shannon-McMillan-Breiman Theorem. Part II (Chapters 8-13) shifts focus to continuous maps of metric spaces, exploring the collection of invariant measures corresponding to a given map. Part III (Chapters 14-16) presents advanced topics rarely found in textbooks at this level, including SRB measures, their deep connection to entropy and Lyapunov exponents, and extensions to two important settings: random and infinite-dimensional dynamical systems. Throughout, the authors emphasize not only the mathematical elegance of ergodic theory but also its practical relevance and rich connections to other areas of mathematics, from information theory to stochastic processes

Contents:

Measure-preserving transformations

Three basic concepts: recurrence, ergodicity and isomorphisms

Ergodic theorems

A hierarchy of mixing properties

Operations on measure-preserving transformations

Entropy

The Shannon-McMillan-Breiman Theorem

Invariant measures for continuous maps

Topological dynamics

Lyapunov exponents

Ingredients in the proof of the Multiplicative Ergodic Theorem

Differentiable maps and invariant densities

Linear operators associated to dynamical systems

Smooth ergodic theory

Random dynamical systems

Infinite-dimensional dynamical systems

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed June 5, 2026)

ISBN:

9783032088369

3032088364

OCLC:

1593911705

Access Restriction:

Restricted for use by site license