Gibbs measures on Subshifts

TL;DR

This paper investigates the conditions under which different notions of Gibbs measures coincide on subshifts, particularly focusing on subshifts of finite type, and characterizes these measures using DLR equations.

Contribution

It proves that for subshifts of finite type, equilibrium measures are also Gibbs measures and provides a characterization using DLR equations, linking different definitions.

Findings

Equilibrium measures on SFTs are Gibbs measures.

Gibbs measures can be characterized via DLR equations on SFTs.

The work bridges definitions of Gibbs measures across mathematical physics and dynamical systems.

Abstract

The notion of Gibbs Measure is used by many researchers of the communities of Mathematical Physics, Probability, Thermodynamic Formalism, Symbolic Dynamics, and others. A natural question is when these several different notions of Gibbs measure coincide. We study the properties of Gibbs measures for functions with $d-$summable variation defined on a subshift $X$. Based on Meyerovitch's work, we prove that if $X$ is a subshift of finite type (SFT), then any equilibrium measure is also a Gibbs measure. Although the definition provided by Meyerovitch does not make any mention to conditional expectations, we show that in the case where $X$ is a SFT, it is possible to characterize these measures in terms of more familiar notions presented in the literature of Mathematical Physics using DLR equations.