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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.
Table des matières
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.
A propos de l'auteur
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).
Résumé
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.
