Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras

TL;DR

This paper establishes a connection between continuous orbit equivalence of one-sided topological Markov shifts and the flow equivalence of their two-sided counterparts, linking algebraic invariants and Cuntz-Krieger algebra isomorphisms.

Contribution

It proves that continuous orbit equivalence of one-sided shifts implies flow equivalence of two-sided shifts and characterizes this equivalence via Cuntz-Krieger algebra isomorphisms.

Findings

Continuous orbit equivalence implies flow equivalence.

Flow equivalence leads to equal determinants det(id-A).

Cuntz-Krieger algebra isomorphism characterizes continuous orbit equivalence.

Abstract

Let A,B be square irreducible matrices with entries in {0,1}. We will show that if the one-sided topological Markov shifts (X_A,\sigma_A) and (X_B,\sigma_B) are continuously orbit equivalent, then the two-sided topological Markov shifts (\bar X_A,\bar\sigma_A) and (\bar X_B,\bar\sigma_B) are flow equivalent, and hence det(id-A)=det(id-B). As a result, the one-sided topological Markov shifts (X_A,\sigma_A) and (X_B,\sigma_B) are continuously orbit equivalent if and only if the Cuntz-Krieger algebras O_A and O_B are isomorphic and det(id-A)=det(id-B).