Gibbs measures on compact ultra metric spaces

TL;DR

This paper establishes the equivalence between Gibbs measures and Gibbs conformal measures for dynamical systems on compact ultrametric spaces, and applies this to beta-shifts to identify equilibrium measures as Gibbs measures.

Contribution

It proves the equivalence of Gibbs and Gibbs conformal measures for expansive actions on ultrametric spaces and characterizes equilibrium measures for beta-shifts as Gibbs measures.

Findings

Gibbs measure and Gibbs conformal measure are equivalent under specified conditions.

Unique equilibrium measure for functions of summable variation on beta-shifts is a Gibbs measure.

The results apply to dynamical systems with countably infinite groups acting expansively.

Abstract

One proves the equivalence of a Gibbs measure and a Gibbs conformal measure for a dynamical system (G,X) when G is a countably infinite discrete group acting expansively on a compact ultrametric space X. As an application one proves for any beta-shift, that the unique equilibrium measure for a function of summable variation is a Gibbs measure.