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Quasi-discrete spectrum and the Wiener-Wintner ergodic theorem for polynomials. (Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes.) (French) Zbl 0803.47013

Let (Ω,μ) be a separable probability space, T:ΩΩ be measure-preserving. Fix fL 1 . The ergodic theorem of Wiener-Wintner asserts that for μ-a.e. ω,

1 n k=0 n-1 exp(ikα)·f(T k ω)convergesforallα·

The author uses the Van der Corput inequality to show the following generalization: For μ-a.e. ω,

1 n k=0 n-1 φ(P(k))·f(T k ω)converges

for all real polynomials P and all continuous periodic functions φ on . If, in addition, T is weakly mixing and fdμ=0, then for μ-a.e. ω, for all M and all continuous periodic φ,

sup P 1 n k=0 n-1 φ(P(k))·f(T k ω)0,

where the supremum is taken over all real polynomials of degree bounded by M.

The proof of the Wiener-Wintner theorem is based on an equivalence which the author generalizes in the following way: Let E 0 be the set of eigenvalues of T (as an operator on L 2 ) and, for m1, set E m :={fL 2 :|f|1,fT·f ¯E m-1 }. Suppose the linear span of m E m is dense in L 2 , fL 1 and M1. Then fgdμ=0 holds for all gE M if and only if for μ-a.e. ω, for all real polynomials of degree less than or equal to M and every continuous periodic φ,

1 n k=0 n-1 φ(P(k))·f(T k ω)0·

.


MSC:
47A35Ergodic theory of linear operators
28D05Measure-preserving transformations