Almost everywhere convergence of ergodic series
Résumé
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).