Spectral metric spaces for Gibbs measures

TL;DR

This paper constructs spectral metric spaces for Gibbs measures on subshifts, linking noncommutative geometry with thermodynamic formalism, and shows that Gibbs measures can be recovered from spectral data.

Contribution

It introduces a method to build spectral triples for Gibbs measures on subshifts, connecting spectral geometry with measure-theoretic entropy.

Findings

Connes' pseudo-metric is a true metric on the state space.

Gibbs measures are recoverable from noncommutative integration.

Noncommutative volume constant equals reciprocal entropy.

Abstract

We construct spectral metric spaces for Gibbs measures on a one-sided topologically exact subshift of finite type. That is, for a given Gibbs measure we construct a spectral triple and show that Connes' corresponding pseudo-metric is a metric and that its metric topology agrees with the weak-*-topology on the state space over the set of continuous functions defined on the subshift. Moreover, we show that each Gibbs measure can be fully recovered from the noncommutative integration theory and that the noncommutative volume constant of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of the shift invariant Gibbs measure.