Ergodic Theory of Expanding Thurston Maps

  • Zhiqiang Li

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

Table of contents

  1. Front Matter
    Pages i-xii
  2. Zhiqiang Li
    Pages 1-13
  3. Zhiqiang Li
    Pages 15-28
  4. Zhiqiang Li
    Pages 29-35
  5. Zhiqiang Li
    Pages 37-77
  6. Zhiqiang Li
    Pages 79-136
  7. Zhiqiang Li
    Pages 137-160
  8. Zhiqiang Li
    Pages 161-172
  9. Back Matter
    Pages 173-182

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


Thurston Map Postcritically-finite Map Nonuniformly Expanding Map Thermodynamical Formalism Equilibrium State

Authors and affiliations

  • Zhiqiang Li
    • 1
  1. 1.Institute for Mathematical SciencesStony Brook University Institute for Mathematical SciencesStony BrookUSA

Bibliographic information