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Equilibrium states in ergodic theory. (English) Zbl 0896.28006

London Mathematical Society Student Texts. 42. Cambridge: Cambridge University Press. ix, 178 p. (1998).
This is a very well written, self-contained and enjoyable book to read. The goal of this book is to provide an introduction to the theory of equilibrium states and pressure. It consists of six chapters in which the first three give all the necessary theory needed for the theory of equilibrium states and pressure.
In Chapter 1, the author gives the motivation for the theory developed in later chapters. Here, equilibrium states and pressure on finite probability spaces are introduced together with the variational principle. Further, Gibbs measures, large deviation estimates, the Ising model on a finite lattice, equilibrium states for Markovian systems, and absolutely continuous invariant measures are discussed in an elementary setting.
In Chapter 2, basic ergodic theory is discussed and all results and set-ups are done for \(\mathbb{Z}^d\)-actions. What is interesting is that the \(\mathbb{Z}^d\)-version of the ‘relatively recent’ proof of the ergodic theorem (by Kamae and Katznelson & Weiss) is given.
Chapter 3 is devoted to entropy theory for \(\mathbb{Z}^d\)-actions and the Shannon-McMillan-Breiman theorem is proved for the case \(d=1\) and \(d\geq 2\).
Chapter 4 forms the core of this book. Here the pressure \(P(\psi)\) of a semi-continuous function (local energy function) is defined from a variational point of view, namely \[ P(\psi)= \sup_{\mu\in {\mathcal M} (T)} (h(\mu) +u(\psi)), \] where \({\mathcal M} (T)\) is the set of all invariant measures for the \(\mathbb{Z}^d\)-action \(T\), \(h(\mu)\) the measure theoretic entropy and \(\mu (\psi)= \int\psi d\mu\). Equilibrium states are thus defined as measures for which this supremum is achieved. Further, continuity properties of the entropy function and the convex geometrical meaning of equilibrium states as derivatives of the pressure function are discussed; the latter makes it possible to characterize equilibrium states for expansive actions.
Chapter 5 is devoted to Gibbs measures of shift spaces over the \(d\)-dimensional integer lattice. Here, two basic examples are given; (i) Gibbs measures as stationary measures for finite Markov shifts; (ii) phase transition in the 2-dimensional Ising model.
Chapter 6 deals with piecewise differentiable dynamical systems. The connection of Sinai-Bowen-Ruelle measures (SBR) with absolutely continuous measures (w.r.t. Lebesgue measure) is given. This chapter ends by showing that for conformal iterated function systems, invariant measures of maximum Hausdorff dimension can be identified as certain equilibrium states.

MSC:

28D15 General groups of measure-preserving transformations
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
37A99 Ergodic theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
28D20 Entropy and other invariants
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