Ergodic Properties of Certain Strongly Transitive CA with a Continuum of Two-Periodic Points

In our previous papers we present a construction of certain one-sided cellular automata (CA) in a metric Cantor space Ã^M ( M = N or M = Z). We proved that the obtained CA are not positively expansive, but similarly to positively expansive CA in Ã^N, they are open, topologically mixing and strongly transitive, and their topological entropies form the set H ={log n: n ∈ N \{0, 1}}. We also proved that, as in the case of positively expansive CA in Ã^N, the uniform Bernoulli measure is an invariant Borel probability measure of maximal entropy for aforementioned CA. In this sense, our results extend those for positively expansive CA on Ã^N in [F. Blanchard and A. Maass, Israel Journal of Mathematics 99 (1997)] and [M. Boyle, D. Fiebig and U. Fiebig, Journal für die Reine und Angewandte Mathematik 487 (1997)]. In this paper we prove that our CA are also strongly mixing with respect to the Borel uniform Bernoulli measure. The first of the aforementioned papers focuses on ergodic properties of positively expansive CA in Ã^N. Therefore, our new result can be viewed as a significant answer to questions posed by Blanchard and Maass in Section 5 of that work.