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

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

2008-09-03

arxiv.org/abs/0809.0656v2

Mathematics

Dynamical Systems

35 pages v2: minor modifications, clarified title

Scientific paper

We investigate limit theorems for Birkhoff sums of locally H\"older 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.

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