Density of periodic points, invariant measures and almost equicontinuous points of Cellular Automata

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of such automata converges in Cesaro mean to an invariant measure μc. If the initial measure μ is a Bernouilli measure, we prove that the Cesaro mean limit measure μc is shift mixing. Therefore we also show that for any shift ergodic and F -invariant measure μ, the existence of μ-almost equicontinuous points implies that the set of periodic points is dense in the topological support S(μ) of the invariant measure μ. Finally we give a non trivial example of a couple (μ-equicontinuous cellular automata F , shift ergodic and F -invariant measure μ) which has no equicontinuous point in S(μ).