Caractérisation spectrale des systèmes dynamiques du type Wiener–Wintner

Abstract

Let (X, B,μ,T) be an ergodic dynamical system on the finite measure space (X, B,μ,T) and K its Kronecker factor. We give a spectral characterization of the Wiener–Wintner functions we introduced in [1] in terms of the a.e. continuity of the fractional rotated ergodic Hilbert transform. We introduce and study weak Wiener–Wintner functions and weak Wiener–Wintner dynamical systems. We define the logarithmic and power capacities of U, the restriction of T onto K ⊥ and we show that weak Wiener–Wintner dynamical systems are characterized spectrally by these capacities. The study of the L 2 case leads to new classes of dynamical systems.