Cesàro mean distribution of group automata starting from measures with summable decay

Consider a finite Abelian group (G,+), with |G| = p , p a prime number, and φ : G → G the cellular automaton given by (φx)n = μxn + νxn+1 for any n ∈ N, where μ and ν are integers coprime to p. We prove that if P is a translation invariant probability measure onG determining a chain with complete connections and summable decay of correlations, then for any w = (wi : i < 0) the Cesàro mean distribution MPw = lim M→∞ 1 M M−1 ∑ m=0 Pw ◦ φ−m, where Pw is the measure induced by P on G conditioned by w, exists and satisfies MPw = λ , the uniform product measure on G . The proof uses a regeneration representation of P.