Limit behaviour of $\mu -$equicontinuous cellular automata

The concept of $\mu -$equicontinuity was introduced by Gilman to classify cellular automata. We show that under some conditions the sequence of Cesaro averages of a measure $\mu ,$ converge under the actions of a $\mu -$equicontinuous CA. We address questions raised by Blanchard-Tisseur 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.