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Wiener-Wintner dynamical systems. (English) Zbl 1128.37300
Summary: Let $(X,\mathcal{B},\mu, T)$ be an ergodic dynamical system on the finite measure space $(X,\mathcal{B},\mu )$. Let $\mathcal{K}$ denote the Kronecker factor of $T$, i.e. the closed linear span in $L^2$ of the eigenfunctions for $T$. We say that $(X,\mathcal{B},\mu ,T)$ is a Wiener-Wintner (WW) dynamical system of power type $\alpha$ in $L^1$ if there exists in $\mathcal{K}^{\bot}$ a dense set of functions $f$ for which the following holds: there exists a finite positive constant $C_f$ such that $$\left\Vert\sup_{\varepsilon} \bigg|\frac{1}{N} \sum_{n=1}^Nf\circ T^n e^{2\pi in\varepsilon}\bigg|\right\Vert_1\leq \frac{C_f}{N^{\alpha}}$$ for all positive integers $N$. Examples of ergodic dynamical systems with this WW property include $K$ automorphisms as well as some skew products over irrational rotations. For WW dynamical systems a simpler proof of the almost everywhere double recurrence property, random weights with a break of duality can be obtained. They also provide naturally almost everywhere continuous random Fourier series related to the spectral measure of the transformation.

37A05Measure-preserving transformations
28D05Measure-preserving transformations
37A25Ergodicity, mixing, rates of mixing
47A35Ergodic theory of linear operators
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