Graph directed Markov systems: geometry and dynamics of limit sets

R. Daniel Mauldin, Mariusz Urbanski0521825385, 9780521825382

The main focus of this book is on the development of the theory of Graph Directed Markov Systems. This far-reaching generalization of the theory of conformal iterated systems can be applied in many situations, including the theory of dynamical systems. Dan Mauldin and Mariusz Urbanski include much of the necessary background material to increase the appeal of this book to graduate students as well as researchers. They also include an extensive list of references for further reading.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 7
Introduction……Page 9
1 Preliminaries……Page 15
2 Symbolic Dynamics……Page 18
2.1 Topological pressure and variational principles……Page 19
2.2 Gibbs states, equilibrium states and potentials……Page 26
2.3 Perron–Frobenius operator……Page 40
2.4 Ionescu-Tulcea and Marinescu inequality……Page 45
2.5 Stochastic laws……Page 54
2.6 Analytic properties of pressure and the Perron–Frobenius operator……Page 57
2.7 The existence of eigenmeasures of the conjugate Perron–Frobenius operator and of Gibbs states……Page 62
3.1 Summable Hölder families……Page 68
3.2 F-conformal measures……Page 71
4.1 Some properties of conformal maps in………Page 76
4.2 Conformal measures; Hausdorff and box dimensions……Page 85
4.3 Strongly regular, hereditarily regular and irregular systems……Page 101
4.4 Dimensions of measures……Page 104
4.5 Hausdorff, packing and Lebesgue measures……Page 108
4.6 Porosity of limit sets……Page 117
4.7 The associated iterated function system……Page 121
4.8 Refined geometry, F-conformal measures versus Hausdorff measures……Page 123
4.9 Multifractal analysis……Page 137
5.1 Examples of GDMSs in other fields of mathematics……Page 150
5.2 Examples with special geometric features……Page 153
6.1 The Radon-Nikodym derivative………Page 158
6.2 Rate of approximation of the Hausdorff dimension by finite subsystems……Page 167
6.3 Uniform perfectness……Page 170
6.4 Geometric rigidity……Page 174
6.5 Refined geometric rigidity……Page 179
7.1 General results……Page 185
7.2 One-dimensional systems……Page 190
7.3 Two-dimensional systems……Page 195
7.4 Rigidity in dimension………Page 209
8.1 Preliminaries……Page 223
8.2 Topological pressure and associated parameters……Page 226
8.3 Perron–Frobenius operator, semiconformal measures and Hausdorff dimension……Page 232
8.4 The associated hyperbolic system. Conformal and invariant measures……Page 236
8.5 Examples……Page 248
9.1 Preliminaries……Page 252
9.2 The Case………Page 254
9.3 The plane case, d = 2……Page 260
9.4 Proofs of the main theorems……Page 269
Appendix 1 Ergodic theory……Page 276
Appendix 2 Geometric measure theory……Page 278
Glossary of Notation……Page 283
Bibliography……Page 286
Index……Page 294