Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

Sébastien Gouëzel (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L^2 observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.

Domaines: Systèmes dynamiques [math.DS]

Sébastien Gouëzel. Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps. Israel Journal of Mathematics, 2010, 180 (1), pp.1-41. ⟨10.1007/s11856-010-0092-z⟩. ⟨hal-00318082v2⟩