On computing the entropy of cellular automata

We study the topological entropy of a particular class of dynamical systems: cellular automata. The topological entropy of a dynamical system (X,F) is a measure of the complexity of the dynamics of F over the space X. The problem of computing (or even approximating) the topological entropy of a given cellular automata is algorithmically undecidable (Ergodic Theory Dynamical Systems 12 (1992) 255). In this paper, we show how to compute the entropy of two important classes of cellular automata namely, linear and positively expansive cellular automata. In particular, we prove a closed formula for the topological entropy of D-dimensional (D greater than or equal to 1) linear cellular automata over the ring Z(m)(m greater than or equal to 2) and we provide an algorithm for computing the topological entropy of positively expansive cellular automata. (C) 2002 Elsevier Science B.V. All rights reserved.