Some properties of cellular automata with equicontinuity points

We investigate topological and ergodic properties of cellular automata having equicontinuity points. In this class surjectivity on a transitive SFT implies existence of a dense set of periodic points. Our main result is that under the action of such an automaton any shift–ergodic measure converges in Cesàro mean, assuming equicontinuity points have measure 1; the limit measure is described by a formula and some of the properties of its topological support are given.