Limit behaviour of μ-equicontinuous cellular automata

The concept of μ-equicontinuity was introduced in [12] to classify cellular automata. We show that under some conditions the sequence of Cesaro averages of a measure μ, converge under the actions of a μ-equicontinuous CA. We address questions raised in [3] on whether the limit measure is either shift-ergodic, a uniform Bernoulli measure or ergodic with respect to the CA. Many of our results hold for CA on multidimensional subshifts.