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

Details Cesaro mean distribution of group automata starting from measures with summable decay Cesaro mean distribution of group automata starting from measures with summable decay

1999-12-16

arxiv.org/abs/math/9912135v1

Ergodic Theory and Dynamical Systems (2000)

Mathematics

Probability

Scientific paper

10.1017/S0143385700000924

Consider a finite Abelian group (G,+), with |G|=p^r, p a prime number, and F: G^N -> G^N the cellular automaton given by {F(x)}_n= A x_n + B x_{n+1} for any n in N, where A and B are integers relatively primes 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 w= (w_i:i<0) the Cesaro mean distribution of the time iterates of the automaton with initial distribution P_w --the law P conditioned to w on the left of the origin-- converges to the uniform product measure on G^N. The proof uses a regeneration representation of P.

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