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

We will study circle actions on Cuntz–Krieger algebras trivially acting on its canonical maximal abelian C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.