Asymptotic randomization of sofic shifts by linear cellular automata

Abstract

Abstract. Let bM/b = bZ/bsupiD/i/sup be a iD/i-dimensional lattice, and let (biA/i/b, +) be an abelian group. biA/isupM/sup/b is then a compact abelian group under componentwise addition. A continuous function Φ : biA/isupM/sup/b → biA/isupM/sup/b is called ia linear cellular automaton/i if there is a finite subset bF/b ⊂ bM/b and non-zero coefficients φf ∈ iZ/i so that, for any ba/b ∈ biA/isupM/sup/b, Φ(ba/b) = Σsubf∈bF/b/subφf · σsupf/sup(ba/b). Suppose that iµ/i is a probability measure on biA/isupM/sup/b whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on Φ and iµ/i) under which Φ iasymptotically randomizes µ/i, meaning that wk* − limsubbJ/b∋ij,/i→∞/sub Φisupj/supµ/i = iη/i, where iη/i is the Haar measure on biA/isupM/sup/b, and bJ/b ⊂ bN/b has Cesàro density one. In the case when Φ = 1 + σ and biA/i/b = (bZ/bsub/p/sub)sups/sup (ip/i prime), we provide a condition on iµ/i that is both necessary and sufficient. We then use this to construct zero-entropy measures which are randomized by 1 + σ.