Cesaro limit and ergodicity of locally eventually periodic measures

We study translation invariant deterministic dynamics (phi) on the lattice (cellular automata). In particular the evolution and limit of probability measures that give the set of locally eventually phi-periodic points full measure. We prove the convergence of the mean averages under phi of this measures. We characterize the ergodicity of the limit measures (solving a question posed by Blanchard and Tisseur) and we prove that in the limit phi is a mixture measure theoretical odometers.