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Convergence of weighted polynomial multiple ergodic averages. (English) Zbl 1175.37003
Let (Y,ν) be a probability space with measure ν, and let {b n } be a bounded sequence of real numbers b n (nZ). The sequence {b n } is called to be universally good for the convergence in the mean of polynomial multiple ergodic averages (in short, for c.m.p.m.e.a.), if for any system (X,μ,T), for any r1, all polynomials with integer coefficients p 1 ,p 2 ,,p r and all f 1 ,,f r L (μ) the averages (1/N) n=0 N-1 b n T p 1 (n) f 1 T p r (n) f r converges in L 2 (μ) as N. The author proves the following theorem: Let (Y,ν,S) be an ergodic system and φL (ν). Then there exists Y 0 Y with ν(Y 0 )=1 such that for every y 0 Y 0 the sequence b n =φ(S n y 0 ) (nZ) is universally good for c.m.p.m.e.a.¬†. By making use of the generalized Wiener-Wintner theorem proved by Host and Kra, the author proves the main theorem by proving the convergence criterion for weighted averages.
MSC:
37A05Measure-preserving transformations
37A30Ergodic theorems, spectral theory, Markov operators