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 hμ (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∞.

Keywords

• Cellular Automata
• Measure Entropy
• Topological Entropy