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

Abstract

Let $B$, $B$ 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 ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ are isomorphic and $det(id-A)=det(id-B)$.