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By Wieslaw Tadeusz Zelazko

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1 (X, A, µ, T) and (X, A, µ) a A dynamical system is defined as the set is a mesure preserving transformation In this chapter and unless otherwise specified the measure of µ(X) = 1 . 2 Consider two dynamical systems X will be (X, A, µ, T) The product of these dynamical systems is the system and (X © Y, A @ B, µ © v, T x S) . A B It is simple to check that the product of two dynamical systems is also a dynamical system ; the measure of the rectangles x is preserved under T x S. Let us consider the product of the dynamical systems ('lI', B ('lI' ), m, Re.

Definition 2 . 6 For an ergodic measure preserving system (X , A , µ , T ), K is called the Kronecker factor of the system. The Kronecker factor can be characterized spectrally. We assume that the reader is familiar with the spectral theorem for normal operators. ) references [Katznelson (1 968 )], [Nadkarni (1 9 9 9 )] and [Queffelec (1 987 )]. The spectral theorem gives us the existence of a spectral measure on

Definition 2. 7 A funtion f E L 1 is said to satisfy the Wiener Wintner property if there exists a set Xf of full measure such that for each x E X f exists for each t E R. 6) . We skip its proof and leave it as an exercise. 1 Let (X, A , µ , T) be a measure preserving dynamical system and f E L 1 (µ) . Assume that there exists a sequence of functions fn E L 1 converging to f in L 1 norm and satisfying the Wiener Wintner property. Then f satisfies the Wiener Wintner property. 3 Wiener Wintner theorem through the affinity of measures Wiener Wintner theorem through the affinity of measures 29 Let us see first what is the idea in following this path of the affinity principle.

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