SOME NONROBUST BERNOULLI-SHIFT RULES

Abstract

In this paper, it is shown that elementary cellular automata rule 172, as a member of the Chua's robust period-1 rules and the Wolfram class I, is also a nonrobust Bernoulli-shift rule. This rule actually exhibits complex Bernoulli-shift dynamics in the bi-infinite binary sequence space. More precisely, in this paper, it is rigorously proved that rule 172 is topologically mixing and has positive topological entropy on a subsystem. Hence, rule 172 is chaotic in the sense of both Li–Yorke and Devaney. The method developed in this paper is also applicable to checking the subshifts contained in other robust period-1 rules, for example, rules 168 and 40, which also represent nonrobust Bernoulli-shift dynamics.