Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps
Abstract
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 L2 observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.
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