## Equilibrium States in Ergodic TheoryThis 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 |

169 | |

List of special notations | 174 |

175 | |

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### Common terms and phrases

a-algebra absolutely continuous assumption Bernoulli measure Birkhoff's ergodic theorem Borel cr-algebra circle maps configuration constant convergence theorem Corollary countable cylinder sets defined definition denote disjoint dp(x dynamical systems elementary energy function ip entropy function equilibrium ergodic decomposition estimate Example Exercise Let exists fibred system finite partition follows G C(X G ES(ip g G G G M(T Gibbs measures GS(ip HD(FQ Hence hT(p inequality invariant measures ip G USC(X Ising model Jensen's inequality large deviations lattice Lebesgue measure lim sup lim^oo Markov maximal measurable space metric space monotone notation Observe p-partition p(ip particular points probability measure probability space probability vector properties prove ps(ip regular local energy Remark Section 1.1 sequence shift spaces sub-a-algebra subset Suppose T-invariant theory topology unique upper semicontinuous variational principle