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- An Introduction to Ergodic Theory.
- Ergodic Theory.
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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.
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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. Help Centre.
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The authors deserve special kudos for their collection of over exercises, many with hints and solutions at the end of the book. I hope a newer edition will highlight more such material that could help a neophyte reader connect with the latest advances and guide future research directions. As a first step, having a webpage dedicated to such additions to the text would certainly enhance the material and make it more attractive and alive for students. This is followed by a few key examples: decimal expansions, continued fractions via the Gauss map, toral rotations, and conservative flows.
The chapter ends with a description of using first-return maps to construct induced systems, and a brief section on multiple recurrence theorems. The second chapter, Existence of invariant measures , proves the fundamental existence result that guarantees a Borel probability measure invariant under the action of a continuous map on a compact metric space. The authors give three approaches to prove this theorem, the slickest of which is an application of the non-trivial Schauder-Tychonoff fixed-point theorem.
They also present the more intuitive and elegant averaging argument that is usually credited to N. Kryloff and N. This is followed by various characterizations of ergodicity, and a useful loopback to the examples discussed in the first chapter with a view to proving ergodicity.
There is a brief discussion of the ergodic measures forming the extremal points of the convex set of invariant probability measures for a fixed dynamical system, which could have been moved to the start of the next short chapter on the Ergodic decomposition. The authors prove the fundamental ergodic decomposition theorem which guarantees the representation of an invariant measure as an integral convex combination of the extremal ergodic measures via the Rokhlin disintegration theorem. The authors take the pedagogically sound strategy of presenting more than a single proof of fundamental results, when available.
The sixth chapter, Unique ergodicity , describes systems that admit exactly one invariant probability measure. On the positive side, they prove that every transitive translation on a compact metrizable topological group is uniquely ergodic with the Haar measure as the unique invariant probability.
Ergodic Theory Course | Omri Sarig
The seventh chapter, Correlations , studies the evolution of the correlations between observables which could measure temperature, or spatial position via a characteristic function as time tends to infinity. This begins with the notions of strong and weak mixing, and their characterization via the Koopman operator. The authors then move to a section on finite-memory processes, a. Markov shifts , which generalize the class of Bernoulli shifts which model I.
An IET is a bijection of a compact interval with a finite number of discontinuities and whose restriction to every interval of continuity is a translation. However, they fail to mention that it was only a decade ago that Avila-Forni proved that a typical IET is either weakly mixing or it is an irrational rotation, see Weak Mixing for Interval Exchange Transformations and Translation Flows , Ann. The eighth chapter introduces the famous isomorphism problem that aims to classify ergodic systems and motivates the next chapter on the seminal concept of Kolmogorov-Sinai, a.
The definition of entropy is highly non-trivial and the authors spend the next few sections on theorems that help calculate entropy in pleasant circumstances, via the Kolmogorov-Sinai and Shannon-McMillan-Breiman theorems. The reader is then returned to computations in the realms of Markov shifts, the Gauss map and linear endomorphisms of tori. The tenth chapter, Variational principle , introduces the notion of topological entropy and its generalization known as pressure , and is concerned with a careful proof that the topological entropy of a continuous map acting on a compact metric space equals the supremum of measure-theoretic entropies with respect of all invariant probability measures.
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The focus is on the specific, but fundamental, class of expanding dynamical systems. Notes in Mathematics vol. Preface 1.
Recurrence 2. Existence of invariant measures 3. Ergodic theorems 4. Ergodicity 5. Ergodic decomposition 6. Unique ergodicity 7. Correlations 8.