Strictly Temporally Periodic Points in Cellular Automata

TL;DR

This paper investigates the properties of strictly temporally periodic points in surjective cellular automata, revealing their density or emptiness depending on the automaton's dynamical characteristics, with specific results for additive automata.

Contribution

It characterizes the density and emptiness of strictly periodic points in various classes of cellular automata, linking these properties to dynamical features like equicontinuity and transitivity.

Findings

Strictly periodic points are dense in almost equicontinuous surjective automata.

Strictly periodic points are absent in positively expansive automata.

In additive automata, the set of strictly periodic points is either dense or empty depending on topological transitivity.

Abstract

We study the set of strictly periodic points in surjective cellular automata, i.e., the set of those configurations which are temporally periodic for a given automaton but they not spatially periodic. This set turns out to be dense for almost equicontinuous surjective cellular automata while it is empty for the positively expansive ones. In the class of additive cellular automata, the set of strictly periodic points can be either dense or empty. The latter happens if and only if the cellular automaton is topologically transitive.