Extensions of Certain Transitive CA with Finite Sets of m−periodic Points
Janusz Matyja
It is suggested in a literature of the subject to undertake a research on transitivity of cellular automata CA and its stronger variants in the metric Cantor space B
(B
) as well as on possibilities of classification of transitive CA in this space up to topological conjugacy. In the response to these suggestions, we present an alternative, simple proof of the strong transitivity of surjective CA in B with memory m > 0 according to the stronger variant of this notion. Then, generalising a known method, we prove that for any integer m > 0 a set of m−periodic points of this type CA is finite. We also prove that extensions, from B to B, of positively expansive CA and constructions of CA which are bijective, expansive and topologically conjugate to two-sided full shifts, also have this property. Obtained in such a way one-sided CA in B and surjective CA in B with memory m > 0 are topologically mixing and they are not topologically conjugate to one-sided open, topologically mixing and strongly transitive CA in B (B) with a continuum of 2−periodic points constructed in our previous papers.
Keywords: Cellular automata, symbolic dynamics, topological conjugacy, strong transitivity, Devaney chaos.