This book presents the interplay between topological Markov shifts and CuntzKrieger 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 CuntzKrieger algebras, and the K-theory dates of the CuntzKrieger 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 CuntzKrieger algebras. The author will also study the relationship between the topological conjugacy of topological Markov shifts and the gauge actions of CuntzKrieger algebras.
