Abstract

We study Cesàro means (time averages) of the evolution measures of the class of permutative cellular automata over {0, 1}? defined by (?BX)n = Nn + R + ?j=0 R 1 (l + bj + Nn + j), where B = b0 ? bR 1 is an aperiodic block in {0, 1}R and operations are taken mod 2. If the initial measure is Bernoulli, we prove that the limit of the Cesàro mean of the first column distribution exists. When R=1 and B=1, ?B is the mod 2 sum automaton. For this automaton we show that the limit is the (1/2, 1/2)-Bernoulli measure, and if the initial measure is Markov, we show that the limit of Cesaro mean of the one-site distribution is equidistributed.