By Idris Assani
The Wiener Wintner ergodic theorem is a strengthening of Birkhoff pointwise ergodic theorem. introduced by way of N. Wiener and A. Wintner, this theorem has brought the learn of a common phenomenon in ergodic thought during which samplings are "good" for an uncountable variety of structures. This publication reviews the speed of convergence within the uniform model of this theorem and what are termed Wiener Wintner dynamical structures and turn out for those structures pointwise effects: the a.e. double recurrence theorem and the a.e. continuity of the fractional circled ergodic Hilbert rework. a few extensions of the Wiener Wintner ergodic theorem also are given.