Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets

Mauldin R.D., Urbanski M.0511070918, 0521825385

The main focus of this book is the exploration of the geometric and dynamic properties of a far reaching generalization of a conformal iterated function system – a Graph Directed Markov System. These systems are very robust in that they apply to many settings that do not fit into the scheme of conformal iterated systems.This book leads readers onto frontier research in the field, making it ideal for both established researchers and graduate students.

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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