# Equilibrium States in Ergodic Theory

Cambridge University Press, Jan 22, 1998 - Mathematics - 178 pages
This book provides a detailed introduction to the ergodic theory of equilibrium states giving equal weight to two of its most important applications, namely to equilibrium statistical mechanics on lattices and to (time discrete) dynamical systems. It starts with a chapter on equilibrium states on finite probability spaces that introduces the main examples for the theory on an elementary level. After two chapters on abstract ergodic theory and entropy, equilibrium states and variational principles on compact metric spaces are introduced, emphasizing their convex geometric interpretation. Stationary Gibbs measures, large deviations, the Ising model with external field, Markov measures, Sinai-Bowen-Ruelle measures for interval maps and dimension maximal measures for iterated function systems are the topics to which the general theory is applied in the last part of the book. The text is self contained except for some measure theoretic prerequisites that are listed (with references to the literature) in an appendix.

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### Contents

 Elementary examples of equilibrium states 1 12 Systems on finite lattices 7 13 Invariant distributions for Markov matrices 12 14 Invariant measures for interval maps 13 Some basic ergodic theory 21 22 Ergodicity and mixing 30 23 The ergodic decomposition 35 24 Return times and return maps 38
 53 Equilibrium states are Gibbs measures 103 54 Markov chains 108 55 Equilibrium states of the Ising model 109 56 Large deviations for Gibbs measures 115 Equilibrium states and derivatives 123 61 SinaiBowenRuelle measures 124 62 Transfer operators 130 63 Absolutely continuous equilibrium states 135

 25 Factors and extensions 40 Entropy 43 32 Entropy of dynamical systems 49 33 Entropy as a function of the measure 58 Equilibrium states and pressure 61 42 Equilibrium states and the entropy function 66 43 Equilibrium states and convex geometry 73 44 The variational principle 78 45 Equilibrium states for expansive actions 89 Gibbs measures 95 52 Gibbs measures are equilibrium states 99
 64 Iterated function systems IFS 145 65 Pressure and dimension for IFS 147 Appendix 157 A3 Nonnegative matrices 158 A4 Some facts from probability and integration 159 A5 Making discontinuous mappings continuous 166 References 169 List of special notations 174 Index 175 Copyright