An Introduction to Ergodic Theory
By addebook • Jul 3rd, 2008 • Category: Mathematics •
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An Introduction to Ergodic Theory
Publisher: Springer
Number Of Pages: 250
Publication Date: 2000-10-06
Sales Rank: 534585
ISBN / ASIN: 0387951520
EAN: 9780387951522
Binding: Paperback
Manufacturer: Springer
Studio: Springer
Average Rating: 4
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.
Review:
A well organized and easy to read introduction
This text helped me understand the fundamentals of ergodic theory and allowed me to see how the subject is organized. The explanations of concepts are fairly brief, but the quality of the mathematical exposition is very good. There are several proofs absent however, but it does not greatly diminish the value of the text. In fact, it keeps the length down, so that it is not such a daunting task to finish the book.
One can cover the basic concepts of ergodic theory with relatively few classes of examples, which include rotations, endomorphisms, and affine transformations of compact groups preserving Haar measure, as well as Bernoulli and Markov shifts.
The preliminary chapter is a quick review of basic measure theory and functional analysis. In the following chapter, ergodicity is described as a form of quantitative recurrence, specifically a measure preserving transformation is ergodic if every set of positive measure A, almost every point of the space eventually gets mapped into A. Later, once the ergodic theorem is proved, this is show to be equivalent to the property that the time average of an integrable function along almost every orbit converges to its integral over the whole space. The last two sections introduce mixing and weak mixing, which give stronger forms of recurrence than ergodicity.
Chapter two introduces basic equivalence relations between measure preserving systems, which include isomorphism, conjugacy, and spectral isomorphism. The following chapter deals with the class of measure-preserving transformations with discrete spectrum, for which the eigenvalues of the associated unitary operator on L^2 completely determine the conjugacy class.
Chapter four develops the fundamentals of entropy theory, taking no shortcuts to this extremely important subject. The entropy of a measure-preserving system is a numerical invariant which quantifies the asymptotic information generated by the system. Two systems which are conjugate are shown to have the same entropy. This was the breakthrough that allowed a proof of what was before the difficult problem of determining, for example whether or not the Bernoulli shifts (1/2,1/2) and (1/3,1/3,1/3) are conjugate, since they have different entropy. The chapter concludes with an introduction to the K-property and the Pinsker partition.
The next part of the book begins with topological dynamics, where recurrence is studied from the continuous perspective. This gives a qualitative as opposed to quantitative description of the long term behavior of the system. It goes on to discuss invariant measures for continuous maps, as well as unique ergodicity. Next we are introduced to the topological counterpart of entropy, which is called topological entropy. It characterizes the exponential growth of the topological complexity of the orbit structure as a single number.
The remainder of the book goes on to develop relationships between topological and measure-theoretic entropy, beginning with the variational principle. This fundamental result states that the topological entropy is the supremum of measure-theoretic entropies, where the supremum is taken over all invariant probability measures. It goes on to discuss measures of maximal entropy, the distribution of periodic points, topological pressure, and equilibrium states.
The last chapter mentions the multiplicative ergodic theorem, which is a fundamental result in the theory of nonuniformly hyperbolic dynamical systems on manifolds. There are a few other modern developments that could have made it into the book, but I think anyone interested enough would be able to find the appropriate literature.
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