Symbolic Dynamics

Abstract

In this chapter, we will study a class of topological dynamical systems known as symbolic dynamical systems. These systems play an important role in coding theory, combinatorial dynamics and theory of cellular automata. In Sect. 2, we introduce the basic concepts associated with such systems. In Sect. 3, we introduce the notion of entropy. In Sect. 4, we compute the measure theoretic entropy of Bernoulli shifts. In Sect. 5, we consider a class of symbolic dynamical systems related to tiling spaces, and prove a result due to M. Szegedy that asserts that any translational tiling of $\mathbb {Z}^{d}$ by a finite set F is periodic when |F| is prime. The last section is devoted to an algebraic dynamical system known as 3-dot system. Using the concept of directional homoclinic groups we show that $\mathbb {Z}^{2}$-actions on symbolic spaces can exhibit strong rigidity property.