Almost everywhere convergence of ergodic series

TL;DR

This paper establishes almost everywhere convergence criteria for ergodic series involving dynamical systems and functions with zero mean, linking convergence to the square-summability of coefficients and exploring spectral properties.

Contribution

It provides necessary and sufficient conditions for convergence of ergodic series and characterizes when the system forms a Riesz basis, with applications to hyperbolic dynamics and Weierstrass functions.

Findings

Series converges almost everywhere if and only if coefficients are square-summable.

The system forms a Riesz basis if spectral measure is absolutely continuous with bounded Radon-Nikodym derivative.

Conditions are verified for Gibbs measures and Hölder functions in hyperbolic systems.

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 $\mu$. 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 $\sum_{n=0}^\infty |a_n|^2<\infty$, 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 $\mu$ relative to hyperbolic dynamics $T$ and for H\"{o}lder functions $f$. An application is given to the study of differentiability of the Weierstrass type functions $\sum_{n=0}^\infty a_n f(3^n x)$.