Chapter 10 Symbolic and algebraic dynamical systems

Abstract

This chapter provides a brief survey of a class of dynamical systems that contains shifts of finite type (both one- and multi-dimensional), as well as actions of one or more commuting automorphisms of a compact Abelian group. The connection between classical (one-dimensional) shifts of finite type and automorphisms of compact Abelian groups via Markov partitions is classical and well understood. Combined with tools from commutative algebra, allows a systematic treatment of ℤ d -actions by the automorphisms of compact Abelian groups and leads to a comprehensive theory of such actions. The wealth of interesting and unexpected properties exhibited by such “algebraic” ℤ d -actions shows that multi-parameter ergodic theory is much more than an elementary extension of the classical theory of ℤ d -actions and flows. The insights into ℤ d -actions gained in the algebraic context has helped to stimulate a number of recent publications on the higher-dimensional shifts of finite type.