Periodic Points and the Measure of Maximal Entropy of an Expanding Thurston Map

TL;DR

This paper investigates the dynamics of expanding Thurston maps, establishing fixed point counts, equidistribution of preimages and periodic points, and the properties of the measure of maximal entropy, including its uniqueness and statistical limits.

Contribution

It provides new results on fixed point enumeration, equidistribution, and measure properties for expanding Thurston maps, linking them to symbolic dynamics and random orbit limits.

Findings

Number of fixed points is 1 + degree of the map.

Preimages and periodic points are equidistributed with respect to the measure of maximal entropy.

The measure of maximal entropy is the weak* limit of atomic measures on random backward orbits.

Abstract

In this paper, we show that each expanding Thurston map $f : S^2\rightarrow S^2$ has $1+ deg f$ fixed points, counted with appropriate weight, where $ deg f$ denotes the topological degree of the map $f$. We then prove the equidistribution of preimages and of (pre)periodic points with respect to the unique measure of maximal entropy $\mu_f$ for $f$. We also show that $(S^2,f,\mu_f)$ is a factor of the left shift on the set of one-sided infinite sequences with its measure of maximal entropy, in the category of measure-preserving dynamical systems. Finally, we prove that $\mu_f$ is almost surely the weak$^*$ limit of atomic probability measures supported on a random backward orbit of an arbitrary point.