Ground states and periodic orbits for expanding Thurston maps

TL;DR

This paper studies invariant measures and ergodic optimization for expanding Thurston maps, establishing new results like the Livšic theorem and the Typically Periodic Optimization Conjecture, including for non-rational examples.

Contribution

It introduces the analysis of maximizing measures and ground states for expanding Thurston maps, extending ergodic optimization results beyond rational maps.

Findings

Verifies the Typically Periodic Optimization Conjecture for generic Hölder potentials.

Shows the existence and uniqueness of maximizing measures supported on periodic orbits.

Applies results to Misiurewicz--Thurston rational maps and Lattès maps.

Abstract

Expanding Thurston maps form a class of branched covering maps on the topological $2$-sphere $S^{2}$, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumption initially investigated by W. P. Thurston, M. Bonk, D. Meyer, P. Ha\"issinsky, and K. M. Pilgrim. The measures of maximal entropy and the absolutely continuous invariant measures for these maps have been studied by these authors, and equilibrium states by the first-named author. In this paper, we initiate the investigation on two new classes of invariant measures, namely, the maximizing measures and ground states, and establish the Liv\v{s}ic theorem, a local Anosov closing lemma, and give a positive answer to the Typically Periodic Optimization Conjecture from ergodic optimization for these maps. As an application, we establish these results for Misiurewicz--Thurston rational maps (i.e., postcritically-finite rational maps without periodic critical points) on the Riemann sphere including the Latt\`es maps with respect to the spherical metric. Our strategy relies on the visual metrics developed by the above authors. In particular, we verify, in a first non-uniformly expanding setting, the Typically Periodic Optimization Conjecture, establishing that for a generic H\"{o}lder continuous potential, there exists a unique maximizing measure, moreover, this measure is supported on a periodic orbit, it satisfies the locking property, and it is the unique ground state. The expanding Thurston maps we consider include those that are not topologically conjugate to rational maps; in particular, they can have periodic critical points.