Gibbs Measures on Multidimensional Spaces. Equivalences and a Groupoid Approach

TL;DR

This paper unifies various definitions of Gibbs measures across different mathematical disciplines for subshifts, especially finite type, and introduces a groupoid approach linking Gibbs measures to KMS states.

Contribution

It proves the equivalence of multiple Gibbs measure definitions for subshifts with summable variation and introduces a groupoid framework connecting Gibbs measures with KMS states.

Findings

All definitions coincide for finite type subshifts.

Established equivalence of Gibbs measures and KMS states via groupoid approach.

Unified framework across dynamical systems, probability, and operator algebras.

Abstract

We consider some of the main notions of Gibbs measures on subshifts introduced by different communities, such as dynamical systems, probability, operator algebras, and mathematical physics. For potentials with $d$-summable variation, we prove that several of the definitions considered by these communities are equivalent. In particular, when the subshift is of finite type (SFT), we show that all definitions coincide. In addition, we introduced a groupoid approach to describe some Gibbs measures, allowing us to show the equivalence between Gibbs measures and KMS states (the quantum analogous to the Gibbs measures).