Weak expansion properties and large deviation principles for expanding Thurston maps

TL;DR

This paper investigates the dynamical properties of expanding Thurston maps, establishing conditions for asymptotic $h$-expansiveness, existence of equilibrium states, and large deviation principles, thereby advancing understanding of their statistical behavior.

Contribution

It characterizes when expanding Thurston maps are asymptotically $h$-expansive and proves the existence of equilibrium states and large deviation principles for certain maps.

Findings

Expanding Thurston map is asymptotically $h$-expansive iff no periodic critical points.

No expanding Thurston map is $h$-expansive.

Existence of equilibrium states and large deviation principles for maps without periodic critical points.

Abstract

In this paper, we prove that an expanding Thurston map $f\colon S^2 \rightarrow S^2$ is asymptotically $h$-expansive if and only if it has no periodic critical points, and that no expanding Thurston map is $h$-expansive. As a consequence, for each expanding Thurston map without periodic critical points and each real-valued continuous potential on $S^2$, there exists at least one equilibrium state. For such maps, we also establish large deviation principles for iterated preimages and periodic points. It follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding Thurston map without periodic critical points and a potential that is H\"older continuous with respect to a visual metric on $S^2$.