Flip conjugacy and asymptotic continuous orbit equivalence of Smale spaces

TL;DR

This paper explores the relationship between flip conjugacy and asymptotic continuous orbit equivalence in Smale spaces, establishing new characterizations using Ruelle algebras and extending to topological Markov shifts.

Contribution

It introduces the concepts of asymptotic topological conjugacy and flip conjugacy in Smale spaces and characterizes them via associated Ruelle algebras and $C^*$-subalgebras.

Findings

Flip conjugacy is characterized by periodic point preserving homeomorphisms.

Asymptotic topological conjugacy is characterized through Ruelle algebras with dual actions.

Flip conjugacy classes of topological Markov shifts are characterized via associated Ruelle algebras.

Abstract

We study asymptotic continuous orbit equivalence of Smale spaces. We prove that two irreducible Smale spaces are flip conjugate if and only if there exists a periodic point preserving homeomorphism giving an asymptotic continuous orbit equivalence between them. We introduce a notion of asymptotic topological conjugacy and asymptotic flip conjugacy in Smale spaces and characterize them in terms of the associated Ruelle algebras with dual actioons. We finally characterize the flip conjugacy classes of irreducible two-sided topological Markov shifts in terms of the associated Ruelle algebras with its $C^*$-subalgebras.