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arXiv · 2310.00415

Wieler solenoids: non-Hausdorff expansiveness, Cuntz-Pimsner models, and functorial properties

Robin J. Deeley·Menevse Eryüzlü·Magnus Goffeng·Allan Yashinski
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Abstract

Building on work of Williams, Wieler proved that every irreducible Smale space with totally disconnected stable sets can be realized via a stationary inverse limit. Using this result, the first and fourth listed authors of the present paper showed that the stable $C^*$-algebra associated to such a Smale space can be obtained from a stationary inductive limit of a Fell algebra. Its spectrum is typically non-Hausdorff and admits a self-map related to the stationary inverse limit. With the goal of understanding the fine structure of the stable algebra and the stable Ruelle algebra, we study said self-map on the spectrum of the Fell algebra as a dynamical system in its own right. Our results can be summarized into the statement that this dynamical system is an expansive, surjective, local homeomorphism of a compact, locally Hausdorff space and from its $K$-theory we can compute $K$-theoretical invariants of the stable and unstable Ruelle algebra of a Smale space with totally disconnected stable sets.

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Robin J. Deeley, Menevse Eryüzlü, Magnus Goffeng, Allan Yashinski. 2023-09-30. Wieler solenoids: non-Hausdorff expansiveness, Cuntz-Pimsner models, and functorial properties. https://arxiv.org/abs/2310.00415

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