Symbolic Dynamics and Matrices

The main purpose of this article is to give some overview of matrix problems and results in symbolic dynamics. The basic connection is that a nonnegative integral matrix A defines a topological dynamical system known as a shift of finite type. Questions about these systems are often equivalent to questions about “persistent” or “asymptotic” aspects of nonnegative matrices. Conversely, tools of symbolic dynamics can be used to address some of these questions. At the very least, the ideas of conjugacy, shift equivalence and strong shift equivalence give viewpoints on nonnegative matrices and directed graphs which are at some point inevitable and basic (although accessible, and even elementary).