By Idris Assani
The Wiener Wintner ergodic theorem is a strengthening of Birkhoff pointwise ergodic theorem. introduced via N. Wiener and A. Wintner, this theorem has brought the examine of a normal phenomenon in ergodic conception during which samplings are "good" for an uncountable variety of structures. This e-book stories the speed of convergence within the uniform model of this theorem and what are termed Wiener Wintner dynamical platforms and end up for those platforms pointwise effects: the a.e. double recurrence theorem and the a.e. continuity of the fractional turned around ergodic Hilbert rework. a few extensions of the Wiener Wintner ergodic theorem also are given.
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Extra info for Wiener Wintner Ergodic Theorems
Example text
4 N --+ oo we obtain the desired inequality. 24) 0 The pointwise ergodic theorem through a variational inequality We already noticed that the first proof of Birkhoff theorem required the knowledge of a dense set of functions on which one could easily obtain the pointwise convergence of the averages. But if we do not have such a simple dense set what can we do? e. a Cauchy sequence. e. Cauchy. We present here such a quadratic variation in the case of the Cesaro averages This application of J . Bourgain's method in the case of the standard ergodic averages is due to H.
Then �; ) ! ( �: ) ! dv . [ ( It can be shown that A(v1 , v2 ) is independent of v and that A(v1 , v2 ) = 0 if and only if v1 and v2 are mutually singular. 2) is given by the next theorem. Theorem 2 . 2 Let a and b be two sequences having correlations. Let us denote by Va and Vb the measures associated with their correlation. Then In parti cular if Va and vb are mutu ally singular then A proof of this result can be found in the original paper of [Coquet et al . (1977)] or in [Queffelec (1987)] .
In other words f satisfies the Wiener Wintner property. 6 we can extend this D Wiener Wintner property to all £ 1 functions. 6 Wiener Wintner Pointwise Ergodic Theorems Topological Wiener Wintner ergodic theorem We have seen that with the added assumption of continuity one could get some partial control of the set of full measure X f (generic points). More control and subsequently sharper results can be expected if the transfor mation T is continuous. We assume that the measures associated with the measure preserving systems are probabilities.