Spectral triples and Gibbs measures for expanding maps on Cantor sets

TL;DR

This paper constructs spectral triples for expanding maps on Cantor sets to recover Gibbs measures as noncommutative measures, linking dynamical systems with noncommutative geometry.

Contribution

It introduces a method to build spectral triples from expanding maps on Cantor sets that encode Gibbs measures as noncommutative measures.

Findings

Spectral triples successfully recover Gibbs measures.

The approach bridges dynamical systems and noncommutative geometry.

Provides a new framework for analyzing expanding maps on fractal sets.

Abstract

Let $T : \Lambda \to \Lambda$ be an expanding map on a Cantor set. For each suitably normalized H\"older continuous potential, we construct a spectral triple from which one may recover the associated Gibbs measure as a noncommutative measure.