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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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 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 |
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Status: | Published |
Refereed: | Yes |
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: | 0143-3857 |
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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