The Wiener Wintner ergodic theorem is a strengthening of Birkhoff pointwise ergodic theorem. Announced by N Wiener and A Wintner, this theorem has introduced the study of a general phenomenon in ergodic theory in which samplings are "good" for an uncountable number of systems. We study the rate of convergence in the uniform version of this theorem and what we call Wiener Wintner dynamical systems and prove for these systems two pointwise results: the a.e. double recurrence theorem and the a.e. continuity of the fractional rotated ergodic Hilbert transform. Some extensions of the Wiener Wintner ergodic theorem are also given.
The Mean and Pointwise Ergodic Theorems; Wiener Wintner Pointwise Ergodic Theorems; Universal Weights for Dynamical Systems; J Bourgain's Return Times Theorem; Extensions of the Return Times Theorem; Speed of Convergence in the Uniform Wiener Wintner Theorem; Weak Wiener Wintner Dynamical Systems; Polynomial Wiener Wintner Ergodic Theorem; Extension to More General Operators.