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 Wider 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 analyse 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 lead to a Cuntz-Pimsner model for the stable Ruelle algebra. The K-theoretic invariants of Cuntz-Pimsner algebras are then studied using the Cuntz-Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo- Martin-Schwinger (KMS) weights on the stable Ruelle algebra of a Wider solenoid. We conclude with several examples of Wider solenoids, their associated algebras and spectral triples.