Prevalence of odometers in cellular automata

Abstract:

We consider left permutive cellular automata $\Phi$ with no memory and positive anticipation, defined on the space of all doubly infinite sequences with entries from a finite alphabet. For each such automaton that is not one-to-one, there is a dense set of points $x$ such that $\Phi : \operatorname {cl} \{\Phi ^n(x) : n \ge 0\} \to \operatorname {cl} \{\Phi ^n(x) : n \ge 0\}$ is topologically conjugate to an odometer, the “$+1$” map on the countable product of finite cyclic groups. This set is a dense $G_\delta$ subset of an appropriate subspace. We identify the odometer in several cases.