Wieler solenoids: non-Hausdorff expansiveness, Cuntz-Pimsner models, and functorial properties

TL;DR

This paper explores the structure of Wieler solenoids and their associated $C^*$-algebras, revealing non-Hausdorff dynamical systems and computing $K$-theoretic invariants for Smale spaces with totally disconnected stable sets.

Contribution

It introduces a detailed analysis of the self-map on the spectrum of the Fell algebra, showing it is an expansive, surjective local homeomorphism on a non-Hausdorff space, and computes related $K$-theoretic invariants.

Findings

The spectrum admits a non-Hausdorff, locally Hausdorff structure.

The self-map is an expansive, surjective local homeomorphism.

$K$-theory computations provide invariants for Ruelle algebras.

Abstract

Building on work of Williams, Wieler proved that every irreducible Smale space with totally disconnected stable sets can be realized via a stationary inverse limit. Using this result, the first and fourth listed authors of the present paper showed that the stable $C^*$-algebra associated to such a Smale space can be obtained from a stationary inductive limit of a Fell algebra. Its spectrum is typically non-Hausdorff and admits a self-map related to the stationary inverse limit. With the goal of understanding the fine structure of the stable algebra and the stable Ruelle algebra, we study said self-map on the spectrum of the Fell algebra as a dynamical system in its own right. Our results can be summarized into the statement that this dynamical system is an expansive, surjective, local homeomorphism of a compact, locally Hausdorff space and from its $K$-theory we can compute $K$-theoretical invariants of the stable and unstable Ruelle algebra of a Smale space with totally disconnected stable sets.