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∞.
Highlights
The additive cellular automata theory and the entropy of this additive cellular automata have grown up somewhat independently, there are strong connections between entropy theory and cellular automata theory.We give an introduction to additive cellular automata theory and discuss the entropy of this additive cellular automata
We show that there is a maximal measure for additive cellular automata
We have found a generating partition for the additive one-dimensional cellular automata f∞ (Lemma)
Summary
The additive cellular automata theory and the entropy of this additive cellular automata have grown up somewhat independently, there are strong connections between entropy theory and cellular automata theory.We give an introduction to additive cellular automata theory and discuss the entropy of this additive cellular automata. [6] Ward has used (n, ε)-separated sets to calculate the topological entropy of cellular automata. We can calculate the topological entropy of additive cellular automata f∞ . We show that there is a maximal measure for additive cellular automata.
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