# Asymptotic continuous orbit equivalence of Smale spaces and Ruelle algebras

## TL;DR

This paper introduces new notions of asymptotic orbit equivalence in Smale spaces, characterizes them via asymptotic Ruelle algebras, and explores their structure through extended Ruelle algebras and Cuntz--Krieger algebras.

## Contribution

It defines asymptotic continuous orbit equivalence and relates it to asymptotic Ruelle algebras, providing a new framework for understanding Smale spaces and their associated $C^*$-algebras.

## Key findings

- Asymptotic Ruelle algebra characterized by dual actions.
- Extended Ruelle algebra as a fixed point algebra under circle action.
- Connection established between asymptotic Ruelle algebra and Cuntz--Krieger algebras.

## 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 $C^*$-algebra which 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.