Limit behaviour of $\mu-$equicontinuous cellular automata

TL;DR

This paper investigates the long-term behavior of $$-equicontinuous cellular automata, showing convergence of Cesaro averages of measures and addressing the nature of limit measures, including ergodicity and Bernoulli properties.

Contribution

It establishes conditions for convergence of Cesaro averages in $$-equicontinuous CA and explores the ergodic properties of the resulting limit measures, extending results to multidimensional subshifts.

Findings

Cesaro averages of measures converge under $$-equicontinuous CA.

Limit measures can be shift-ergodic or Bernoulli, depending on conditions.

Results apply to multidimensional subshifts.

Abstract

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.