Wieler solenoids, Cuntz-Pimsner algebras and K-theory

TL;DR

This paper investigates the K-theoretic invariants of Wieler solenoids, connecting dynamical systems, operator algebras, and K-theory, and provides explicit models and examples for these complex structures.

Contribution

It introduces a Cuntz-Pimsner model for the stable Ruelle algebra of Wieler solenoids and analyzes their K-theoretic invariants using Morita equivalences and spectral triples.

Findings

Characterization of unstable sets as fiber bundles

Explicit Morita equivalence between groupoids

KMS weights on the stable Ruelle algebra

Abstract

We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler's Theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyze an explicit groupoid Morita equivalence between the groupoids of Deaconu-Renault and Putnam-Spielberg, extending results of Thomsen. The Deaconu-Renault groupoid and the explicit Morita equivalence leads to a Cuntz-Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants of Cuntz-Pimsner algebras are then studied using the Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions we characterize the KMS weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.