Strictly Temporally Periodic Points in Cellular Automata

Abstract

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

On One-Sided, D-Chaotic CA Without Fixed Points, Having Continuum of Periodic Points With Period 2 and Topological Entropy log(p) for Any Prime p

“…Now basing on the properties of F for a prime p, we conclude from Theorem 2 in [19] and Theorem 7.10 in [20] that the cartesian product F × σ is topologically conjugated to a cellular automaton (A N n , F n ), which has the expected properties. The obtained results coincide with research directions pointed out by papers [1,2,6,7,11,[13][14][15]18]. …”

“…The dynamics of transitive cellular automata in metric Cantor spaces A N and A Z has been intensively investigated [6][7][8][9]. It has been established that any positively expansive cellular automaton in A N is topologically conjugated with a one-sided, topologically mixing SFT [1,2], is non-injective [1,10,11], E-chaotic [12] and has topological entropy log(n), n ≥ 2, n ∈ N [1].…”