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Ergodic Properties of Certain Strongly Transitive CA with a Continuum of Two-Periodic Points
Janusz Matyja
In our previous papers we present a construction of certain one-sided cellular automata (CA) in a metric Cantor space Γπ (π = β or π = β€). We proved that the obtained CA are not positively expansive, but similarly to positively expansive CA in Γβ, they are open, topologically mixing and strongly transitive, and their topological entropies form the set π» = {log π : π β β \ {0, 1}}. We also proved that, as in the case of positively expansive CA in Γβ, 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 Γβ in [F. Blanchard and A. Maass, Israel Journal of Mathematics 99 (1997)] and [M. Boyle, D. Fiebig and U. Fiebig, Journal fΓΌr 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 Γβ. Therefore, our new result can be viewed as a significant answer to questions posed by Blanchard and Maass in Section 5 of that work.
Keywords: Mixing properties, symbolic dynamics, cellular automata
Mathematics Subject Classification:
Primary 37A25; Secondary 37A35, 37B10, 37B15, 37B40