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Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. (English) Zbl 1080.37003

Consider G is a nilpotent Lie group, X a compact homogeneous space of G and X=G/Γ the nilmanifold on which G acts by left translations. A sequence {g(n)} n in G of the form g(n)=a 1 p 1 (n) a m p m (n) where the a i G and p i are polynomials taking on integer values on the integers is called a polynomial sequence. The author establishes the following pointwise convergence result for continuous functions on such a nilmanifold:

For any xX, for any continuous function fC(X) and for any Folner sequence Φ N in

lim N 1 Φ N nΦ N (g(n)x)

exists. The proof is done by studying carefully the distribution of the orbit of a point on a compact nilmanifold.

This pointwise result and its multidimensional extension obtained by the author [ibid. 25, 215–225 (2005; Zbl 1080.37004)] are useful tools as they allow one to reduce the study of the norm and pointwise convergence of several nonconventional averages to orthocomplements of characteristic factors.


MSC:
37A15General groups of measure-preserving transformation
28D15General groups of measure-preserving transformations
22F30Homogeneous spaces