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Invariant measures for bipermutative cellular automata
1. | Department of Mathematics, Trent University, Peterborough, Ontario, K9L 1Z8, Canada |
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.
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2021 Impact Factor: 1.588
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