On the Measure Entropy of Additive Cellular Automata f_∞

Abstract: We show that for an additive one-dimensional cellular automata f_∞ on space of all doubly infinitive sequences with values in a finite set S = {0, 1, 2, ..., r-1}, determined by an additive automaton rule [equation] (mod r), and a f_∞-invariant uniform Bernoulli measure μ, the measure-theoretic entropy of the additive one-dimensional cellular automata f_∞ with respect to μ is equal to h_μ (f_∞) = 2klog r, where k ≥ 1, r-1∈S. We also show that the uniform Bernoulli measure is a measure of maximal entropy for additive one-dimensional cellular automata f_∞.