Abstract

A Wiener-Wintner Double Recurrence Theorem

Let (X,F, μ, T) be a standard ergodic dynamical system (i.e. X is a compact metric space, μ a Borel probability measure on X, and T a homeomorphism of X). It is known that all Lebesgue dynamical systems are isomorphic to standard ones. Birkhoff’s classical Pointwise Ergodic Theorem states that for μ almost every x, \frac{1}{N} ∑_{n=0}^{N−1} f(T^{n}x) converges. The classical Wiener-Wintener Ergodic Theorem (one proof can be found in [A]), states that for μ almost every x, \frac{1}{N} ∑_{n=0}^{N−1} f(T^{n}x)e^{inθ} converges for every real number θ. The difficulty here is that a single set of full measure works for an uncountable number of θ. Bourgain’s almost everywhere Double Recurrence Theorem [B],which answered a difficult conjecture of Furstenberg, states that for every pair of integers a, b, for every f, g ∈ L^{2}(X, μ), and for μ almost every x, \frac{1}{N} ∑_{n=0}^{N−1} f(T^{an}x)g(T^{bn}x) converges. One of the main tools used by Bourgain in his proof, is a uniform version of the Wiener-Winter Ergodic Theorem.

Our purpose is to prove the following. Suppose that T is totally ergodic and that a and b are integers, then ∀f, g ∈ L^{2}(μ) there exist a set X̃ of full measure so that ∀x ∈ X̃, \frac{1}{N} ∑_{n=0}^{N−1} f(T^{an}x)g(T^{bn}x)e^{inθ} converges for every real number θ. By taking, f to be the constant one function, we see that this generalizes the Wiener-Wintner Theorem, and taking θ = 0, we see a generalization of Bourgain’s Theorem. Each of these theorems are, in turn, generalizations of Birkhoff’s Theorem.

The techniques employed here, involve decomposing L^{2}(X, μ) into the Conze-Lesigne algebra [CL],[R] (the maximal factor of the type “compact abelian group extension of the Kronecker factor”) and its orthogonal complement. A proof has been obtained in the case where either function is in the orthogonal complement of the Conze-Lesigne algebra, and in the case where both functions are in the Conze-Lesigne algebra assuming that the compact abelian group extension is determined by an affine cocycle. A proof of the general result, employing these special cases, is in progress.

References

[A] I. Assani, “A Wiener-Wintner propery for the helical transform” Ergod. Th. and Dynam. Sys., 1992, 12, 185-194.

[B] J. Bourgain, “Double recurrence and almost sure convergence” Journal für die reine und angewandte Mathematik, 1990, 404, 140-161.

[L] E. Lesigne “Équations fonctionnnelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales” Bull. Soc. Math. France, 1993, 121, 315-351.

[R] D. Rudolph, “Eigenfunctions of T × S and the Conze-Lesigne Algebra” Proceedings of the Alexandria Conference, 1993.