Nuclear dimension and classification of C*-algebras associated to Smale spaces

TL;DR

This paper proves that homoclinic C*-algebras of mixing Smale spaces are classifiable by their Elliott invariant, establishing finite nuclear dimension using dynamic asymptotic dimension.

Contribution

It demonstrates finite nuclear dimension for these C*-algebras and classifies them via the Elliott invariant, advancing understanding of their structure.

Findings

Homoclinic C*-algebras are classifiable by the Elliott invariant.

Stable, unstable, and homoclinic C*-algebras have finite nuclear dimension.

Application of dynamic asymptotic dimension to these algebras.

Abstract

We show that the homoclinic C*-algebras of mixing Smale spaces are classifiable by the Elliott invariant. To obtain this result, we prove that the stable, unstable, and homoclinic C*-algebras associated to such Smale spaces have finite nuclear dimension. Our proof of finite nuclear dimension relies on Guentner, Willett, and Yu's notion of dynamic asymptotic dimension.