Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules (Q1431407)

A cellular automaton (c.a.) \(f:A^\mathbb{Z}\to A^\mathbb{Z}\) \((A\) a finite set) is defined by a block map \(\varphi:A^n\to A\) via \((f(x))_i=\varphi (x_{i+l},\dots, x_{i+r})\) \((r-l=n-1)\). The first results give necessary conditions for \(f\) to be conjugate through a one block map to a group translation c.a. (i.e., the alphabet has the form \(G\times B\) where \(G\) is a group and the block maps are \(\varphi(g,h)=gh\) for \(G\) and \(n=r=l= 1\) for \(B)\) and to an affine translation c.a. (similarly defined as before). Section 4 deals with invariant measures. It is shown that for certain c.a. \(\lambda_K^\mathbb{Z}\) is the unique invariant measure where \(\lambda_K\) is the Haar measure on the group \(K=\mathbb{Z}_p\), \(p\) prime. Finally, in section 5 a variety of results for c.a. are presented which ensure harmonic mixing for translation invariant measures. This is applied to show Césaro convergence to product measures for affine translation c.a. [cf. \textit{D. A. Lind}, Cellular automata, Proc. Interdisc. Workshop, Los Alamos/N. M. 1983, 36--44 (1984; Zbl 0562.68038)].