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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 $X=G/{\Gamma }$ the nilmanifold on which $G$ acts by left translations. A sequence ${\left\{g\left(n\right)\right\}}_{n\in ℤ}$ in $G$ of the form $g\left(n\right)={a}_{1}^{{p}_{1}\left(n\right)}\cdots {a}_{m}^{{p}_{m}\left(n\right)}$ where the ${a}_{i}\in G$ 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 $x\in X,$ for any continuous function $f\in C\left(X\right)$ and for any Folner sequence ${{\Phi }}_{N}$ in $ℤ$

$\underset{N}{lim}\frac{1}{{{\Phi }}_{N}}\sum _{n\in {{\Phi }}_{N}}\left(g\left(n\right)x\right)$

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:
 37A15 General groups of measure-preserving transformation 28D15 General groups of measure-preserving transformations 22F30 Homogeneous spaces
##### Keywords:
pointwise convergence; nilmanifold; polynomial sequences