Extensions of one-sided surjective CA with certain measure-theoretic properties

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 M with the Borel uniform Bernoulli measure µ = µ|B(B M), where either M = ℕ or M = ℤ. We prove that extensions, from B N to B ℤ, of surjective CA which are strongly mixing with respect to µ|B(B ℕ), have the same property with respect to µ|B(B ℤ). Additionally, we present an alternative, general and simple justification that extensions, from B ℕ to B ℤ, of surjective CA for which µ|B(B N) is a measure of maximal entropy, have the same property with respect to µ|B(B ℤ). As a consequence, one-sided and non positively expansive CA which are extensions, from B N to B ℤ, of positively expansive ones, are strongly mixing with respect to µ|B(B ℤ) which is a measure of maximal entropy.