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 (XA,σA) and (XB,σB) are continuously orbit equivalent, then the two-sided topological Markov shifts (X‾A,σ‾A) and (X‾B,σ‾B) are flow equivalent, and hence det(id−A)=det(id−B). As a result, the one-sided topological Markov shifts (XA,σA) and (XB,σB) are continuously orbit equivalent if and only if the Cuntz–Krieger algebras OA and OB are isomorphic and det(id−A)=det(id−B).