Ergodic Theory of Expanding Thurston Maps (Atlantis Studies in Dynamical Systems (4))

Thurston maps are topological generalizations of postcritically-finite rational maps. This book provides a comprehensive study of ergodic theory of expanding Thurston maps, focusing on the measure of maximal entropy, as well as a more general class of invariant measures, called equilibrium states, and certain weak expansion properties of such maps. In particular, we present equidistribution results for iterated preimages and periodic points with respect to the unique measure of maximal entropy by investigating the number and locations of fixed points. We then use the thermodynamical formalism to establish the existence, uniqueness, and various other properties of the equilibrium state for a Holder continuous potential on the sphere equipped with a visual metric. After studying some weak expansion properties of such maps, we obtain certain large deviation principles for iterated preimages and periodic points under an additional assumption on the critical orbits of the maps. This enables us to obtain general equidistribution results for such points with respect to the equilibrium states under the same assumption. Preface 7 Contents 9 Notation 11 1 Introduction 13 2 Thurston Maps 26 2.1 Historical Background 27 2.2 Definition for Thurston Maps 28 2.3 Cell Decompositions 29 2.4 Notions of Expansion for Thurston Maps 33 2.5 Visual Metric 34 2.6 Invariant Curves 36 3 Ergodic Theory 40 3.1 Covers and Partitions 40 3.2 Entropy and Pressure 41 3.3 The Ruelle Operator for Expanding Thurston Maps 43 3.4 Weak Expansion Properties 45 4 The Measure of Maximal Entropy 47 4.1 Number and Locations of Fixed Points 50 4.2 Equidistribution 63 4.3 Expanding Thurston Maps as Factors of the Left-Shift 83 5 Equilibrium States 88 5.1 The Assumptions 92 5.2 Existence 93 5.3 Uniqueness 113 5.4 Ergodic Properties 129 5.5 Co-homologous Potentials 133 5.6 Equidistribution 141 5.7 A Random Iteration Algorithm 143 6 Asymptotic h-Expansiveness 146 6.1 Some Properties of Expanding Thurstons Maps 148 6.2 Concepts from Graph Theory 152 6.3 Proof of Theorem 6.1 153 7 Large Deviation Principles 170 7.1 Level-2 Large Deviation Principles 173 7.2 Characterizations of the Pressure P(f,φ) 175 7.3 Proof of Large Deviation Principles 178 7.4 Equidistribution Revisited 179 References 182 Index 187 Front Matter....Pages i-xii Introduction....Pages 1-13 Thurston Maps....Pages 15-28 Ergodic Theory....Pages 29-35 The Measure of Maximal Entropy....Pages 37-77 Equilibrium States....Pages 79-136 Asymptotic h-Expansiveness....Pages 137-160 Large Deviation Principles....Pages 161-172 Back Matter....Pages 173-182