We prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval. As an immediate corollary we obtain a concentration inequality. We apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure.
Consider a finite Abelian group (G, +), with |G| = p r , p a prime number, and ϕ : G N → G N the cellular automaton given by (ϕx) n = µx n +νx n+1 for any n ∈ N, where µ and ν 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 Cesàro mean distribution, where P w is the measure induced by P on G N conditioning to w, exists and satisfies M P w = λ N , the uniform product measure on G N . The proof uses a regeneration representation of P.
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