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Almost everywhere convergence of ergodic series

Published online by Cambridge University Press:  06 October 2015

AIHUA FAN
AIHUA FAN*
Affiliation:
School of Mathematics and Statistics, Central China Normal University, 430079 Wuhan, China and
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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 $\sum _{n=0}^{\infty }a_{n}f(3^{n}x)$ .

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 490 - 511
Copyright
© Cambridge University Press, 2015 

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