Extensions of One-Sided Surjective CA with Certain Measure-Theoretic Properties
Janusz Matyja
It is suggested in the literature of the subject to undertake research on ergodicity and measure-theoretic entropy of surjective cellular automata (CA) in the metric Cantor space B with the Borel uniform Bernoulli measure μ = μ|β(B𝕄), where either 𝕄 = ℕ or 𝕄 = ℤ. We prove that extensions, from Bℕ to Bℤ, of surjective CA which are strongly mixing with respect to μ|β(Bℕ), have the same property with respect to μ|β(Bℤ). Additionally, we present an alternative, general and simple justification that extensions, from Bℕ to Bℤ, of surjective CA for which μ|β(Bℕ) is a measure of maximal entropy, have the same property with respect to μ|β(Bℤ). As a consequence, one-sided and non positively expansive CA which are extensions, from Bℕ to Bℤ, of positively expansive ones, are strongly mixing with respect to μ|β(Bℤ) which is a measure of maximal entropy.
Keywords: Surjective cellular automata, uniform Bernoulli measure, mixing properties, entropy