Equilibrium States in Ergodic Theory

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Cambridge University Press, Jan 22, 1998 - Mathematics - 178 pages
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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
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