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In this chapter, we will show the classification theorem for flow equivalence of two-sided topological Markov shifts. The theorem says that two-sided topological Markov shifts \((\bar{X}_A,\bar{\sigma }_A)\) and \((\bar{X}_B,\bar{\sigma }_B)\) are flow equivalent if and only if there exists an isomorphism \(\bar{\Phi }:{{\mathcal {O}}_A}\otimes {\mathcal {K}}\rightarrow {{\mathcal {O}}_B}\otimes {\mathcal {K}}\) of \(C^*\) -algebras such that \(\bar{\Phi }({{\mathcal {D}}_A}\otimes {\mathcal {C}}) = {{\mathcal {D}}_B}\otimes {\mathcal {C}}\) .

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Classification Theorem for Flow Equivalence and Topological Conjugacy

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  • 发表日期：

    2025年

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  • Kengo Matsumoto

摘要

In this chapter, we will show the classification theorem for flow equivalence of two-sided topological Markov shifts. The theorem says that two-sided topological Markov shifts \((\bar{X}_A,\bar{\sigma }_A)\) and \((\bar{X}_B,\bar{\sigma }_B)\) are flow equivalent if and only if there exists an isomorphism \(\bar{\Phi }:{{\mathcal {O}}_A}\otimes {\mathcal {K}}\rightarrow {{\mathcal {O}}_B}\otimes {\mathcal {K}}\) of \(C^*\) -algebras such that \(\bar{\Phi }({{\mathcal {D}}_A}\otimes {\mathcal {C}}) = {{\mathcal {D}}_B}\otimes {\mathcal {C}}\) .