Invariant measures for bipermutative cellular automata

A right-sided, nearest neighbour cellular automaton (RNNCA) is a continuous transformation$\Phi:\mathcal A^{\mathbb Z} \rightarrow\mathcal A^{\mathbb Z}$ determined by a local rule$\phi:\mathcal A^{\{0,1\}}\rightarrow\mathcal A$ so that, for any $\mathbf a\in\mathcal A^{\mathbb Z}$ and any$z\in\mathbb Z$, $\Phi(\mathbf a)_z = \phi(a_z,a_{z+1})$. We say that$\Phi$ is bipermutative if, for any choice of $a\in\mathcal A$, the map$\mathcal A\ni b \mapsto \phi(a,b)\in\mathcal A$ is bijective, and also, for anychoice of $b\in\mathcal A$, the map $\mathcal A\ni a \mapsto \phi(a,b)\in\mathcal A$ isbijective. We characterize the invariant measures of bipermutative RNNCA.First we introduce the equivalent notion of a quasigroup CA.Then we characterize $\Phi$-invariant measures when $\mathcal A$ is a(nonabelian) group, and $\phi(a,b) = a\cdot b$. Then we show that, if$\Phi$ is any bipermutative RNNCA, and $\mu$ is $\Phi$-invariant, then$\Phi$ must be $\mu$-almost everywhere $K$-to-1, for some constant$K$. We then characterize invariant measures when $\mathcal \mathcal A^{\mathbb Z}$ is agroup shift and $\Phi$ is an endomorphic CA.

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.