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Ergodic averages of commuting transformations with distinct degree polynomial iterates

Abstract : We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)\cdot\ldots\cdot f_\ell(T_\ell^{p_\ell(n)}x)$, where $p_1,\ldots,p_\ell$ are integer polynomials with distinct degrees, and $T_1,\ldots,T_\ell$ are invertible, commuting measure preserving transformations, acting on the same probability space. This establishes several cases of a conjecture of Bergelson and Leibman, that complement the case of linear polynomials, recently established by Tao. Furthermore, we show that, unlike the case of linear polynomials, for polynomials of distinct degrees, the corresponding characteristic factors are mixtures of inverse limits of nilsystems. We use this particular structure, together with some equidistribution results on nilmanifolds, to give an application to multiple recurrence and a corresponding one to combinatorics.
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https://hal.archives-ouvertes.fr/hal-01252587
Contributor : Bernard Host Connect in order to contact the contributor
Submitted on : Thursday, January 7, 2016 - 5:06:43 PM
Last modification on : Monday, April 4, 2022 - 5:25:19 PM

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Qing Chu, Nikos Frantzikinakis, Bernard Host. Ergodic averages of commuting transformations with distinct degree polynomial iterates. Proceedings of the London Mathematical Society, London Mathematical Society, 2011, 102 (5), pp.801-842. ⟨10.1112/plms/pdq037⟩. ⟨hal-01252587⟩

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