Expansivity, Permutivity, and Chaos for Cellular Automata

Abstract.

We study the dynamics of expansive cellular automata. We prove that a cellular automaton is expansive if and only if it is topologically conjugate to an appropriate one-sided subshift. We define a large class of expansive cellular automata in terms of the permutivity of the local rule on which they are based. We prove that each cellular automaton of this class is topologically conjugate to a one-sided full shift. Since one-sided full shifts are widely recognized as the paradigm of chaotic systems, we conclude that the above mentioned cellular automata are chaotic according to any reasonable definition of chaos. We provide a technique to construct expansive cellular automata which are topologically conjugate to one-sided full shifts but do not belong to the above defined class. Finally, we investigate connections between expansivity, topological transitivity, and surjectivity.