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Nonconventional ergodic averages and nilmanifolds. (English) Zbl 1077.37002
The authors prove the following extension of Furstenberg’s theorem [{\it J. Furstenberg}, J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)]: Let $T$ be a measure-preserving invertible transformation on a probability space and $f_1,\dots,f_k$ be bounded measurable functions. Then the limit $$\lim_{n\to\infty}\frac 1n\sum^{n-1}_{i=0}f_1\circ T^i\dots f_k\circ T^{ik}$$ exists in square mean. This answers a long standing question, on which several authors made partial progress in the last 30 years. The authors also prove an extension where averages are taken over cubes of growing size. The proofs use special types of factors, called characteristic.

37A05Measure-preserving transformations
28D05Measure-preserving transformations
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