Summary: Let be an ergodic dynamical system on the finite measure space . Let denote the Kronecker factor of , i.e. the closed linear span in of the eigenfunctions for . We say that is a Wiener-Wintner (WW) dynamical system of power type in if there exists in a dense set of functions for which the following holds: there exists a finite positive constant such that
for all positive integers . Examples of ergodic dynamical systems with this WW property include 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.