Limit Measures for A ne Cellular Automata

Let M be a monoid (e. is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F : A M ?! A M that commutes with all shift maps. Let be a (possibly nonstationary) probability measure on A M ; we develop suucient conditions on and F so that the sequence fF N g 1 N=1 weak*-converges to the Haar measure on A M , in density (and thus, in Cess aro average as well). As an application, we show: if A = Z =p (p prime), F is any \nontrivial" LCA on A (Z D) , and 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 F N weak*-converges to Haar measure in density.