Ergodic theory of discrete dynamical systems (Q2843970)

This is a textbook for a one-semester graduate course on the ergodic theory of discrete dynamical systems. It covers the basic concepts and results of this theory in the context of topological and differentiable discrete dynamics and includes Kingman's subadditivity theorem, a full treatment of Osedelets' multiplicative theorem, the theory of topological and statistical attractors (Chapter 7 covers classical results of Smale and Il'yashenko as well as work on `SRB-like' or `quasiphysical' measures to which the auther herself has contributed), Sinai-Ruelle-Bowen measures, Gibbs measures, Pesin theory, spectral theory, shifts and Bernoulli measures, and metric and topological entropies.NEWLINENEWLINEThe standard of exposition is high, and the book would be suitable for independent study. The reader is assumed to be familiar with the elements of point-set topology, measures and integrals, functional analysis, Fourier analysis and Riemannian geometry. However the author reviews some matters and gives careful statements of all the results used from these areas, and references to sources. All terminology is carefully defined and concepts are motivated. At some points, results are stated and used without proof (notably Rohlin's theorem on measurable decomposition of a probability measure with respect to a measurable partition, the Hirsh-Pugh-Shub theorem on the existence of stable and unstable varieties for uniformly hyperbolic diffeomorphisms, Frank's Lemma, Pesin's entropy formula for \(C^{1+\alpha}\) hyperbolic Gibbs diffeomorphisms in terms of pointwise Lyapunov exponents, the fundamental theorem of Pesin theory on the absolute continuity of local holonomy for \(C^{1+\alpha}\) ergodic hyperbolic diffeomorphisms, the Shadowing Lemma and Bowen's theorem on the local product structure for \(C^1\) Anosov diffeomorphisms), but for the most part complete and meticulous proofs are provided for all the results in the main development, except that small steps are often left to the reader as exercises, with copious hints provided. A few results are proved in less than full generality, where this materially simplifies the exposition. There are many exercises and examples. There are relatively few misprints, and almost none in the first two-thirds of the book. The author has confirmed to the reviewer that in the definitions of Milnor attractor on p. 199 and of Il'yashenko statistical attractor on p. 204, the reader should replace \(m(E_K)=1\) by \(m(E_K)>0\). A small drawback is that the margins are very narrow -- the book is worth a little more paper.NEWLINENEWLINEThe book will be very useful to anyone who seeks precise and complete definitions of widely used terminology in the field. A feature is that the author is careful to restate assumptions often enough so that the reader is kept informed without constantly having to check back. This very readable text would be worth translating into other languages to make it accessible to a wider readership.