Equilibrium States for Expanding Thurston Maps

TL;DR

This paper demonstrates the existence and uniqueness of equilibrium states for expanding Thurston maps using thermodynamical formalism, establishing their ergodic properties and equidistribution results, with applications to rational maps.

Contribution

It introduces a comprehensive thermodynamical framework for expanding Thurston maps, proving uniqueness, ergodicity, and equidistribution of equilibrium states, extending to rational maps on the Riemann sphere.

Findings

Unique equilibrium states exist for each expanding Thurston map with a H"older potential.

The equilibrium states are ergodic, mixing, and exact under the map.

Preimages and backward orbits are equidistributed with respect to these states.

Abstract

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $\mu_\phi$ for each expanding Thurston map $f: S^2\rightarrow S^2$ together with a real-valued H\"older continuous potential $\phi$. Here the sphere $S^2$ is equipped with a natural metric induced by $f$, called a visual metric. We also prove that identical equilibrium states correspond to potentials which are co-homologous upto a constant, and that the measure-preserving transformation $f$ of the probability space $(S^2,\mu_\phi)$ is exact, and in particular, mixing and ergodic. Moreover, we establish versions of equidistribution of preimages under iterates of $f$, and a version of equidistribution of a random backward orbit, with respect to the equilibrium state. As a consequence, all the above results hold for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric.