Symbolics dynamics of elementary cellular automata rule 88

摘要

In this paper, the dynamical behaviors of elementary cellular automata (ECA) rule 88 are studied from the viewpoint of symbolic dynamics. Based on the results derived from the finite case, it is shown that there exist three different Bernoulli-measure subsystems of rule 88 in the space of bi-infinite symbolic sequences. The relationships of these three subsystems and the existence of fixed points are investigated, revealing that the union of them is not the global attractor of rule 88 under the bi-infinite case. Furthermore, the dynamical properties of topologically mixing and topological entropy of rule 88 are exploited on its subsystems. In addition, it is shown that rule 88, a member of Wolfram's class II, possesses richer and more complicated dynamical behaviors in the space of bi-infinite sequences. Finally, it is noted that the method presented in this work is also applicable to study the dynamics of other ECA rules, especially the 112 Bernoulli-shift rules therein.