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Summary: SPECTRAL CHARACTERIZATION OF WIENER WINTNER
DYNAMICAL SYSTEMS
BY I. ASSANI
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 denote by U the restriction of T onto K
the orthocomplement of K. We give a spectral characterization in L2
of Wiener Wintner
functions in terms of the capacity of the support of the maximal spectral type of U and
the a.e. continuity of the fractional rotated ergodic Hilbert transform. The study of the
L2
case leads to new classes of dynamical systems.
1. Introduction
In [A1] we introduced and studied ergodic dynamical systems that we called Wiener
Wintner dynamical systems of power type in Lp. We defined a function f to be a Wiener
Wintner function in the following way:
Definition 1: Let (X, B, µ, T) be an ergodic dynamical system. A function f is a Wiener-
Wintner function of power type in Lp if there exist finite positive constants Cf and
such that
sup
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