On dynamical properties of generalized toggle automata

Abstract

A one-sided one-dimensional cellular automaton F with radius r has associated a canonical factor defined by considering only the first r coordinates of all the images of points under the powers of F. Whenever the cellular automaton is surjective, this language defines a subshift which plays a primary role in its dynamics. In this article we study the class of expansive cellular automata, i.e. those that are conjugate to their canonical factor; they are surjective, and their canonical factor is always of finite type. This class is a natural generalization of the toggle or permutative cellular automata introduced in [H]. We prove that expansive CA are topologically mixing, and their topological entropies are logarithms of integers. If, in addition, we endow these CA with the uniform Bernoulli measure, we prove that it is the unique measure of maximal entropy for F. We also describe a family of non permutative expansive cellular automata.