On the Measure Entropy of Additive Cellular Automata f∞

We show that for an additive one-dimensional cellular automata 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 f(x ∞ f n-k, ..., xn+k) = (mod r), and a -invariant uniform Bernoulli measure μ, the measure-theoretic entropy of the additive one-dimensional cellular automata with respect to μ is equal to h ∑ − = + k i k n x