Linear cellular automata , asymptotic randomization , and entropy

If A = Z/2, then A Z is a compact abelian group. A linear cellular automaton is a shift-commuting endomorphism Φ of A. If μ is a probability measure on A, then Φ asymptotically randomizes μ if Φjμ converges to the Haar measure as j→∞, for j in a subset of Cesàro density one. Via counterexamples, we show that nonzero entropy of μ is neither necessary nor sufficient for asymptotic randomization. If A = Z/2 (with discrete topology), then A Z (with Tychonoff topology) is a compact abelian group. Let σ : A−→A be the shift map (ie. σ(a) = [a′z|z∈Z], where a ′ z = az−1, ∀z ∈ Z). A linear cellular automaton (LCA) is a topological group endomorphism Φ : A−→A that commutes with σ. Let M(A) be the set of Borel probability measures on A, and let η ∈ M(A) be the Haar measure. If μ ∈ M(A), we say that Φ asymptotically randomizes μ if there is subset J ⊂ N of Cesàro density one so that wk− lim J∋j→∞ Φμ = η. LCA randomize a broad class of probability measures, including Bernoulli measures, Markov chains, and Markov random fields [1, 2, 3, 4, 6, 8, 9]. One of the common factors in all these cases is positive entropy. Conversely, randomization is impossible for many zero-entropy measures, such as quasiperiodic or rank one systems [7]. Is positive entropy a necessary/sufficient condition for asymptotic randomization? We will refute both possibilities, by constructing, in §1, a zero-entropy measure which asymptotically randomizes, and in §2, a positive entropy, ergodic measure which doesn’t. Preliminaries: If Φ is any LCA on A, then there is a finite set V ⊂ Z so that Φ can be written as the polynomial of shifts Φ = ∑ v∈V σ . This means, for any a ∈ A, that Φ(a)z = ∑ v∈V az+v for all z ∈ Z. The advantage of this notation is that composition of LCA