Bratteli-Vershik models for zero-dimensional systems (Recent Developments in Dynamical Systems and their Applications)

Abstract

Herman, Putnam and Skau (R. H. Herman, I. F. Putnam and C. F. Skau (1992)) showed some correspondence between the pointed topological conjugacy classes of essentially minimal compact zero-dimensional systems (χ, φ, y) and the equivalence classes of essentially simple ordered Bratteli diagrams. In fact, using these, they made deep investigations into C*-algebraic theories. Later Medynets (K. Medynets (2006)) showed that every Cantor aperiodic system is homeomorphic to the Vershik map acting on the space of infinite paths of an ordered Bratteli diagram. He produced an equivalent class of ordered Bratteli diagrams from a topologically conjugacy classes of a triple (χ, φ, B), in which B is a particular closed set that is called a basic set. In this manuscript, we explain some basic concepts that can extend some topology of their works to the case in which there may be a lot of periodic orbits. In doing this, the work by Downarowicz and Karpel (T. Downarowicz and 0. Karpel(2019)) plays an important role.