The Measure of Maximal Entropy

Abstract

Chapter 4 is devoted to the study of periodic points and the measure of maximal entropy for an expanding Thurston map. In Sect. 4.1, we investigate the fixed points, periodic points, and preperiodic points of the expanding Thurston maps. In particular, we study the location of periodic points and establish a formula for the number of fixed points of each expanding Thurston map (Theorem 4.1). In Sect. 4.2, we prove a number of equidistribution results for periodic points and iterated preimages with respect to the measure of maximal entropy using the exact combinatorial information we obtained in Sect. 4.1. In Sect. 4.3, we show that for each expanding Thurston map f with its measure of maximal entropy \(\mu _f\), the measure-preserving dynamical system \((S^2, f,\mu _f)\) is a factor, in the category of measure-preserving dynamical systems, of the measure-preserving dynamical system of the left-shift operator on the one-sided infinite sequences of \(\deg f\) symbols together with its measure of maximal entropy.