On the measure entropy of additive cellular automata \(f_\infty\)
Summary: We show that for an additive one-dimensional cellular automaton \(f_\infty\) on the space of all doubly infinitive sequences with values in a finite set \(S = \{0, 1, 2,\dots, r-1\}\), determined by an additive automaton rule \(f(x_{n-k},\dots, x_{n+k}) =\sum^k_{i=-k} x_{n+k} (\text{mod } r)\), and an \(f_\infty\) -invariant uniform Bernoulli measure \(\mu\), the measure-theoretic entropy of the additive one-dimensional cellular automaton \(f_\infty\) with respect to \(\mu\) is equal to \(h_\mu (f_\infty) = 2k\log r\), where \(k\geq 1, r-1\in S\) . We also show that the uniform Bernoulli measure is a measure of maximal entropy for additive one-dimensional cellular automata \(f_\infty\).
scientific article; zbMATH DE number 2148942