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Almost everywhere convergence of ergodic series

Abstract : We consider ergodic series of the form Sigma(infinity)(n=0) a(n) f(T-n x), where f is an integrable function with zero mean value with respect to a T-invariant measure mu. Under certain conditions on the dynamical system T, the invariant measure mu and the function f, we prove that the series converges mu-almost everywhere if and only if Sigma(infinity)(n=0) vertical bar a(n)vertical bar(2) < infinity, and that in this case the sum of the convergent series is exponentially integrable and satisfies a Khintchine-type inequality. We also prove that the system \f o T-n\ is a Riesz system if and only if the spectral measure of f is absolutely continuous with respect to the Lebesgue measure and the Radon-Nikodym derivative is bounded from above as well as from below by a constant. We check the conditions for Gibbs measures mu relative to hyperbolic dynamics T and for Holder functions f. An application is given to the study of differentiability of the Weierstrass-type functions Sigma(infinity)(n=0) a(n) f(3(n) x).
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Soumis le : lundi 28 mars 2022 - 16:23:24
Dernière modification le : mercredi 14 septembre 2022 - 17:43:01

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Aihua Fan. Almost everywhere convergence of ergodic series. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2017, 37 (2), pp.490-511. ⟨10.1017/etds.2015.58⟩. ⟨hal-03622007⟩



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