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Ergodic Theory of Expanding Thurston Maps

  • Book
  • © 2017

Overview

Authors:
  1. Zhiqiang Li

    1. Institute for Mathematical Sciences, Stony Brook University Institute for Mathematical Sciences, Stony Brook, USA

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  • A comprehensive study of ergodic theory of expanding Thurston maps
  • The proofs are written with a non-specialist audience in mind
  • Develop thermodynamical formalism for a new class of maps
  • Includes supplementary material: sn.pub/extras

Part of the book series: Atlantis Studies in Dynamical Systems (ASDS, volume 4)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Zhiqiang Li
    Pages 1-13

  3. Thurston Maps

    • Zhiqiang Li
    Pages 15-28

  4. Ergodic Theory

    • Zhiqiang Li
    Pages 29-35

  5. The Measure of Maximal Entropy

    • Zhiqiang Li
    Pages 37-77

  6. Equilibrium States

    • Zhiqiang Li
    Pages 79-136

  7. Asymptotic h-Expansiveness

    • Zhiqiang Li
    Pages 137-160

  8. Large Deviation Principles

    • Zhiqiang Li
    Pages 161-172

  9. Back Matter

    Pages 173-182
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Keywords

About this book

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 enablesus to obtain general equidistribution results for such points with respect to the equilibrium states under the same assumption.

Authors and Affiliations

  • Institute for Mathematical Sciences, Stony Brook University Institute for Mathematical Sciences, Stony Brook, USA

    Zhiqiang Li

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