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

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Let A,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 (Xₐ,σₐ) and (XB,σB) 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).