Symbolic Dynamical Systems and C*-Algebras: Continuous Orbit Equivalence of Topological Markov Shifts and Cuntz–Krieger Algebras

This  book presents the interplay between topological Markov shifts and  Cuntz-Krieger algebras by providing notations, techniques, and ideas in  detail. The main goal of this book is to provide a detailed proof of a  classification theorem for continuous orbit equivalence of one-sided  topological Markov shifts. The continuous orbit equivalence of one-sided  topological Markov shifts is classified in terms of several different  mathematical objects: the étale groupoids, the actions of the continuous  full groups on the Markov shifts, the algebraic type of continuous full  groups, the Cuntz-Krieger algebras, and the K-theory dates of the  Cuntz-Krieger algebras. This classification result shows that  topological Markov shifts have deep connections with not only operator  algebras but also groupoid theory, infinite non-amenable groups, group  actions, graph theory, linear algebras, K-theory, and so on. By using  this classification result, the complete classification of flow  equivalence in two-sided topological Markov shifts is described in terms  of Cuntz-Krieger algebras. The authors will also study the relationship  between the topological conjugacy of topological Markov shifts and the  gauge actions of Cuntz-Krieger algebras.