Spectral triples for hyperbolic dynamical systems

TL;DR

This paper constructs spectral triples for C*-algebras of hyperbolic dynamical systems called Smale spaces, demonstrating that the spectral dimension captures the topological entropy of these systems.

Contribution

It introduces a novel spectral triple framework for Smale spaces and links spectral dimension to topological entropy, advancing the understanding of noncommutative geometry in dynamical systems.

Findings

Spectral triples are successfully constructed for Smale spaces.

Spectral dimension equals the topological entropy of the system.

Provides a new geometric perspective on hyperbolic dynamics.

Abstract

Spectral triples are defined for C*-algebras associated with hyperbolic dynamical systems known as Smale spaces. The spectral dimension of one of these spectral triples is shown to recover the topological entropy of the Smale space.