Linear cellular automata, asymptotic randomization, and entropy
Abstract
If A=Z/2, then AZ is a compact abelian group. A `linear cellular automaton' is a shift-commuting endomorphism F of AZ. If P is a probability measure on AZ, then F `asymptotically randomizes' P if Fj P converges to the Haar measure as j-->oo, for j in a subset of Cesaro density one. Via counterexamples, we show that nonzero entropy of P is neither necessary nor sufficient for asymptotic randomization.
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