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.

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