Ergodic Theory

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 17) lays the foundation, covering invariant measures, measure-theoretic isomorphisms, ergodicity, mixing, entropy, and culminating in the ShannonMcMillanBreiman Theorem.
• Part II (Chapters 813) shifts focus to continuous maps of metric spaces, exploring the collection of invariant measures corresponding to a given map.
• Part III (Chapters 1416) 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.

Starts from the basics and emphasizes material in ergodic theory most relevant to a wide audience Explains coverage of extensions of classical theory to random and infinite-dimensional dynamical systems Provides treatment of physical/observable invariant measures, a vital concept in applications

Autorentext

Lai-Sang Young is a Professor of Mathematics at New York University and the Moses Professor of Science. Born in Hong Kong, she is an American mathematician whose work spans dynamical systems theory, mathematical physics, and computational neuroscience. Her recent honors include delivering a plenary lecture at the International Congress of Mathematicians (2018), election to the U.S. National Academy of Sciences (2020), the SIAM Juergen Moser Award for nonlinear sciences (2021), and the Rolf Schock Prize in Mathematics (2024). Alex Blumenthal is an Associate Professor of Mathematics at the Georgia Institute of Technology. An American mathematician, he works at the interface of ergodic theory, random dynamical systems, and fluid mechanics. His recent honors include an NSF CAREER Award (2023–2028) and a Sloan Research Fellowship (2024).

Inhalt

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.