Pointwise characteristic factors for Wiener Wintner double recurrence theorem

2014/02/27 by Assani, Idris, Duncan, David, Moore, Ryo · 1 citation
#37A05 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1402.7094

Abstract

In this paper, we extend Bourgain's double recurrence result to the Wiener-Wintner averages. Let (X, F, μ, T) be a standard ergodic system. We will show that for any f₁, f₂ ∈ L^∞(X), the double recurrence Wiener-Wintner average (1)/(N) ∑_n=1^N f₁(T^anx)f₂(T^bnx) e^{2πi n t} converges off a single null set of X independent of t as N → ∞. Furthermore, we will show a uniform Wiener-Wintner double recurrence result: If either f₁ or f₂ belongs to the orthogonal complement of the Conze-Lesigne factor, then there exists a set of full measure such that the supremum on t of the absolute value of the averages above converges to 0.

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