Almost everywhere convergence of ergodic series

Abstract

We consider ergodic series of the form $\sum _{n=0}^{\infty }a_{n}f(T^{n}x)$, where $f$ is an integrable function with zero mean value with respect to a $T$-invariant measure $\unicode[STIX]{x1D707}$. Under certain conditions on the dynamical system $T$, the invariant measure $\unicode[STIX]{x1D707}$ and the function $f$, we prove that the series converges $\unicode[STIX]{x1D707}$-almost everywhere if and only if $\sum _{n=0}^{\infty }|a_{n}|^{2}<\infty$, 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 $\{f\circ T^{n}\}$ 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 $\unicode[STIX]{x1D707}$ relative to hyperbolic dynamics $T$ and for Hölder functions $f$. An application is given to the study of differentiability of the Weierstrass-type functions $\sum _{n=0}^{\infty }a_{n}f(3^{n}x)$.