Equilibrium States in Ergodic Theory

by Gerhard Keller

• ISBN13: 9780521594202
• ISBN10: 0521594200
• Format: Hardcover
• Copyright: 1998-02-28
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	vii
1 Elementary examples of equilibrium states	1	(20)
1.1 Equilibrium states in finite systems	1	(6)
1.2 Systems on finite lattices	7	(5)
1.3 Invariant distributions for Markov matrices	12	(1)
1.4 Invariant measures for interval maps	13	(8)
2 Some basic ergodic theory	21	(22)
2.1 Birkhoff's ergodic theorem	21	(9)
2.2 Ergodicity and mixing	30	(5)
2.3 The ergodic decomposition	35	(3)
2.4 Return times and return maps	38	(2)
2.5 Factors and extensions	40	(3)
3 Entropy	43	(18)
3.1 Information and entropy of partitions	43	(6)
3.2 Entropy of dynamical systems	49	(9)
3.3 Entropy as a function of the measure	58	(3)
4 Equilibrium states and pressure	61	(34)
4.1 Pressure	61	(5)
4.2 Equilibrium states and the entropy function	66	(7)
4.3 Equilibrium states and convex geometry	73	(5)
4.4 The variational principle	78	(11)
4.5 Equilibrium states for expansive actions	89	(6)
5 Gibbs measures	95	(28)
5.1 Regular local energy functions	95	(4)
5.2 Gibbs measures are equilibrium states	99	(4)
5.3 Equilibrium states are Gibbs measures	103	(5)
5.4 Markov chains	108	(1)
5.5 Equilibrium states of the Ising model	109	(6)
5.6 Large deviations for Gibbs measures	115	(8)
6 Equilibrium states and derivatives	123	(34)
6.1 Sinai-Bowen-Ruelle measures	124	(6)
6.2 Transfer operators	130	(5)
6.3 Absolutely continuous equilibrium states	135	(10)
6.4 Iterated function systems (IFS)	145	(2)
6.5 Pressure and dimension for IFS	147	(10)
A Appendix	157	(12)
A.1 Lipschitz-continuous functions	157	(1)
A.2 Some convex analysis in R^d	157	(1)
A.3 Non-negative matrices	158	(1)
A.4 Some facts from probability and integration	159	(7)
A.5 Making discontinuous mappings continuous	166	(3)
References	169	(5)
List of special notations	174	(1)
Index	175