Hausdorff Measures and KMS States

TL;DR

This paper establishes a connection between Hausdorff measures and KMS states on certain C*-algebras derived from dynamical systems, showing that the Hausdorff dimension determines the inverse temperature of the KMS state.

Contribution

It introduces a method to construct KMS states from Hausdorff measures for systems with local scaling properties, and proves their uniqueness under mild conditions.

Findings

Hausdorff measure induces a KMS state with inverse temperature equal to Hausdorff dimension

Uniqueness of the KMS state under mild hypotheses

Application to Cuntz, graph, and fractafold C*-algebras

Abstract

Given a compact metric space $X$ and a local homeomorphism $T:X\to X$ satisfying a local scaling property, we show that the Hausdorff measure on $X$ gives rise to a KMS state on the $C^{*}$-algebra naturally associated to the pair $(X,T)$ such that the inverse temperature coincides with the Hausdorff dimension. We prove that the KMS state is unique under some mild hypothesis. We use our results to describe KMS states on Cuntz algebras, graph algebras, and $C^{*}$-algebras on fractafolds.