Topological Chaos of Cellular Automata Rules

Abstract

The dynamical behaviors of elementary cellular automata rules are investigated from the viewpoint of symbolic dynamics in the space of bi-infinite symbolic sequences. It turns out the topologically conjugate equivalences between rules by applying blocking transformation and releasing transformation. Based on this result, the topological chaos of rule 22 is detailedly characterized; that is, rule 22 is topologically mixing and possesses the positive topological entropy on two subsystems. Thus, rule 22 is chaotic in the sense of both Li-Yorke and Devaney on these two subsystems. Finally, it is worth mentioning that the method presented in this paper is also applicable to other blocking transformation equivalences therein.