Published online by Cambridge University Press: 06 October 2015
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
$\sum _{n=0}^{\infty }a_{n}f(3^{n}x)$.