Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

Giovanni Forni; Adam Kanigowski

doi:10.5802/jep.111

Giovanni Forni ¹; Adam Kanigowski ¹

¹ Department of Mathematics, University of Maryland 4176 Campus Drive – William E. Kirwan Hall, College Park, MD 20742-4015, USA

Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 63-91.

Abstract
Résumé

We study time changes of bounded type Heisenberg nilflows $(ϕ_{t})$ acting on the Heisenberg nilmanifold $M$ . We show that for every positive $τ \in W^{s} (M)$ , $s > 7 / 2$ , every non-trivial time change $(ϕ_{t}^{τ})$ enjoys the Ratner property. As a consequence, every mixing time change is mixing of all orders. Moreover, we show that for every $τ \in W^{s} (M)$ , $s > 9 / 2$ and every $p, q \in ℕ$ , $p \neq q$ , $(ϕ_{p t}^{τ})$ and $(ϕ_{q t}^{τ})$ are disjoint. As a consequence, Sarnak conjecture on Möbius disjointness holds for all such time changes.

Nous étudions les reparamétrisations $(ϕ_{t}^{τ})$ des flots nilpotents de Heisenberg de type borné sur une variété nilpotente de Heisenberg $M$ . Nous montrons que, pour des fonctions positives $τ \in W^{s} (M)$ (espace de Sobolev) avec $s > 7 / 2$ , toute reparamétrisation non triviale $(ϕ_{t}^{τ})$ a la propriété de Ratner. En conséquence, toute reparamétrisation mélangeante est mélangeante de tous les ordres. De plus, nous montrons que pour toutes les fonctions $τ \in W^{s} (M)$ , avec $s > 9 / 2$ et pour tous $p, q \in ℕ$ , $p \neq q$ , les flots $(ϕ_{p t}^{τ})$ et $(ϕ_{q t}^{τ})$ sont disjoints. Il s’ensuit, en particulier, que la conjecture de Sarnak sur la disjonction de la fonction de Möbius est valable pour toutes ces reparamétrisations.

Received: 2019-02-17
Accepted: 2019-10-09
Published online: 2019-11-08

MR: 4033750 | Zbl: 07128377

DOI: 10.5802/jep.111
Classification: 37C40, 28D10
Keywords: Nilflows, time-changes, Ratner property, multiple mixing, disjointness
Author's affiliations:

Giovanni Forni ¹; Adam Kanigowski ¹

¹ Department of Mathematics, University of Maryland 4176 Campus Drive – William E. Kirwan Hall, College Park, MD 20742-4015, USA

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     title = {Multiple mixing and disjointness for time changes of bounded-type {Heisenberg} nilflows},
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     pages = {63--91},
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Giovanni Forni; Adam Kanigowski. Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 63-91. doi : 10.5802/jep.111. https://jep.centre-mersenne.org/articles/10.5802/jep.111/

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