Topological and Measure-Theoretic Properties of Certain Open, Topologically Mixing and Strongly Transitive CA
Janusz Matyja
In this paper we strengthen our previous results related to a construction of some type of one-sided cellular automata〈CA〉 in the metric Cantor space Ãℕ(˜Ãℤ). We proved that the mentioned CA have the following properties. They are not positively expansive and like in positively expansive case in Ãℕ, they are right-closing, open, topologically mixing, strongly transitive, and their topological entropies form the set H = {log n : n ≥ 2, n ∈ ℕ}. Additionally, they show a significant divergence in the range of an another invariant of topological conjugacy, i.e., the cardinality of the set of m−periodic points. Their uniqueness lies also in the fact that they have continuum of 2−periodic points. Consequently, in this paper, we justify that they are not topologically conjugate to a wider set of subclasses of transitive CA in Ãℕ(˜Ãℤ) which are often viewed as ergodic or suspected to be ergodic systems with respect to the Borel uniform Bernoulli measure. We also prove that like in positively expansive case in Ãℕ(˜Ãℤ), the mentioned probability measure is a measure of maximal entropy for these CA. It broadens an answer to the question raised by F. Blanchard, A. Maass 〈1997〉 Is it possible to extend some of the results obtained for positively expansive CA in Ãℕ to a certain subclass of right-closing and non positively expansive CA in this space?
Keywords: Cellular automata, symbolic dynamics, measure of maximal entropy, topological conjugacy.