Wieler solenoids, Cuntz-Pimsner algebras and K-theory

Deeley, R., Goffeng, M., Mesland, B. and Whittaker, M. F. (2018) Wieler solenoids, Cuntz-Pimsner algebras and K-theory. Ergodic Theory and Dynamical Systems, 38(8), pp. 2942-2988. (doi: 10.1017/etds.2017.10)

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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 Cuntz-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.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Whittaker, Professor Mike
Authors: Deeley, R., Goffeng, M., Mesland, B., and Whittaker, M. F.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics
Journal Name:Ergodic Theory and Dynamical Systems
Publisher:Cambridge University Press
ISSN (Online):1469-4417
Published Online:02 May 2017
Copyright Holders:Copyright © 2017 Cambridge University Press
First Published:First published in Ergodic Theory and Dynamical Systems 38(8):2942-2988
Publisher Policy:Reproduced in accordance with the publisher copyright policy

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