Wiener–Wintner dynamical systems

Abstract

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 L² 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 L¹ 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 \[\bigg\| \sup_{\varepsilon} \bigg|\frac{1}{N} \sum_{n=1}^Nf\circ T^n e^{2\pi in\varepsilon}\bigg| \bigg\|_1\leq \frac{C_f}{N^{\alpha}}\] 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.