Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz--Krieger algebras

We will study torus actions on Cuntz–Krieger algebras trivially acting on its canonical maximal abelian $C^{\ast }$-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.