Limit Measures for Affine Cellular Automata

Let M be a monoid (e.g. N, Z, or Z), and A an abelian group. A is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F : A −→ A that commutes with all shift maps. Let μ be a (possibly nonstationary) probability measure on A; we develop sufficient conditions on μ and F so that the sequence {Fμ}N=1 weak*-converges to the Haar measure on A, in density (and thus, in Cesàro average as well). As an application, we show: if A = Z/p (p prime), F is any “nontrivial” LCA on A( D), and μ belongs to a broad class of measures (including most Bernoulli measures (for D ≥ 1) and “fully supported” N-step Markov measures (when D = 1), then Fμ weak*-converges to Haar measure in density. This research partially supported by NSERC Canada.