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Invariant measures for bipermutative cellular automata

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  • 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 any choice of $b\in\mathcal A$, the map $\mathcal A\ni a \mapsto \phi(a,b)\in\mathcal A$ is bijective.
    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 a group shift and $\Phi$ is an endomorphic CA.
    Mathematics Subject Classification: Primary: 37B15; Secondary: 37A50.


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