Wieler solenoids, Cuntz-Pimsner algebras and K-theory

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.

References in corpus (3)

• Nonunital spectral triples and metric completeness in unbounded KK-theory
• Shift tail equivalence and an unbounded representative of the Cuntz-Pimsner extension
• The Cuntz-Pimsner extension and mapping cone exact sequences

Cited by in corpus (5)

• Shift tail equivalence and an unbounded representative of the Cuntz-Pimsner extension
• Constructing KMS states from infinite-dimensional spectral triples
• Untwisting twisted spectral triples
• A geometric representative for the fundamental class in KK-duality of Smale spaces
• Spectral triples on irreversible $C^{\ast }$-dynamical systems