Complex Symbolic Dynamics of Bernoulli Shift Cellular Automata Rule

Abstract

In this paper, the complex dynamical behaviors of one dimensional cellular automata rule 11, which is a Bernoulli sigmatau-shift rule, are investigated from the viewpoint of symbolic dynamics. Based on the dynamical properties of subshift of finite type and the relationship between subshift and quasi-subshift, it is strictly proved that rule 11 is topologically mixing on its two subsystems. At the same time, the topological entropies of rule 11 are calculated on the two subsystems, respectively. Conclusively, rule 11 holds rich and complicated dynamical behaviors. For example, it is chaotic in the sense of Li-Yorke and Devaney.