Thermodynamic formalism for coarse expanding dynamical systems

TL;DR

This paper develops a thermodynamic formalism for a broad class of coarse expanding dynamical systems, establishing fundamental statistical properties and equilibrium states, even with complex branch points on the 2-sphere.

Contribution

It generalizes thermodynamic formalism to weakly coarse expanding systems, including those with infinite postcritical sets and on the 2-sphere with repelling branch points.

Findings

Existence and uniqueness of equilibrium states for various potentials.

Proven statistical laws such as CLT and large deviations.

Established exponential decay of correlations.

Abstract

We consider a class of dynamical systems, which we call weakly coarse expanding, which is a generalization to the postcritically infinite case of expanding Thurston maps as discussed by Bonk-Meyer and is closely related to coarse expanding conformal systems as defined by Haissinsky-Pilgrim. We prove existence and uniqueness of equilibrium states for a wide class of potentials, as well as statistical laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and a large deviation principle. Further, if the system is defined on the 2-sphere, we prove all such results even in presence of periodic (repelling) branch points.