Symbolic Dynamics, Group Automorphisms and Markov Partitions

Abstract

These lectures are about symbolic dynamics and the relationship of subshifts of finite type to toral automorphisms and solenoids through topological entropy and Markov partitions. The first part is about subshifts of finite type. Subshifts of finite type have been studied in dynamics and in information theory from a number of different points of view. C. Shannon in 1948 [S] studied them in an information theory context because they model information channels with a finite memory. W. Parry in 1964 [P] investigated them in a topological and ergodic theoretic setting. R. Adler and B. Weiss in 1970 [AW] used them as a combinatorial model for two dimensional toral automorphisms. This allowed a measure theoretic classification of these automorphisms and lead to the use of subshifts of finite type as models for all hyperbolic dynamical systems. In recent years the study of subshifts of finite type has become a subject in its own right. Here, the basic definitions and dynamical properties of these systems are presented. There is also a discussion of topological conjugacy and continuous factor maps. Most of this was known by about ten years ago. Since then there have been new developments but many of the central problems remain unresolved. Some of these problems are discussed here. The next part describes toral automorphisms and solenoids. Toral automorphisms were important in the development of hyperbolic dynamics. An expansive toral automorphism exhibits in a simple, concrete setting the important features of any hyperbolic dynamical system. An expansive toral automorphism has local product structure, its periodic points are dense, its zeta function can be computed, it is structurally stable and it has a kind of algebraic rigidity. In addition, Markov partitions for invertible maps were first constructed on the two dimensional torus. Solenoids are generalizations of toral automorphisms and have all of these properties. The final part of these lectures discusses topological entropy and Markov partitions. Markov partitions tie together the dynamics of subshifts of finite type, toral automorphisms and solenoids. A Markov partition is constructed for a particular two dimensional toral automorphism. It is very easily drawn and exhibits clearly the relationship between subshifts of finite type and toral automorphisms. Finally there is a short general discussion of Markov partitions, their properties and some of their consequences.