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Asymptotic Continuous Orbit Equivalence of Smale Spaces and Ruelle Algebras

Part of: Dynamical systems with hyperbolic behavior Selfadjoint operator algebras

Published online by Cambridge University Press:  09 January 2019

Kengo Matsumoto
Kengo Matsumoto*
Affiliation:
Department of Mathematics, Joetsu University of Education, Joetsu 943-8512, Japan Email: kengo@juen.ac.jp
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Abstract

In the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid $C^{\ast }$-algebra that is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz–Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.

Keywords

hyperbolic dynamicsSmale spaceRuelle algebragroupoidzeta functioncontinuous orbit equivalenceshifts of finite typeCuntz-Krieger algebra

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 46L35: Classifications of $C^*$-algebras
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 5 , October 2019 , pp. 1243 - 1296
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

This work was supported by JSPS KAKENHI Grant Number 15K04896.

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