Abstract

Consider a finite Abelian group (G, +), with -G- = p(r), p a prime number, and phi : G(N) --> G(N) the cellular automaton given by (phix)(n) = mux(n) + nux(n+1) for any n is an element of N where mu and nu are integers coprime to p. We prove that if P is a translation invariant probability measure on G(Z) determining a chain with complete connections and summable decay of correlations, then for any <()under bar> = (omega (i) : i < 0) the Cesaro mean distribution M-P<()under bar> = lim(M-->infinity)1/M Sigma (M-1)(m=0)P(<()under bar>)circle phi (-m) where P-<()under bar> is the measure induced by P on G(N) conditioned by <()under bar>, exists and satisfies M-P<()under bar> = lambda (N) the uniform product measure on G(N). The proof uses a regeneration representation of P.