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

[Mélange multiple et disjonction pour les reparamétrisations des flots nilpotents de type borné]

Forni, Giovanni¹ ; Kanigowski, Adam¹

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, Tome 7 (2020), pp. 63-91.

Abstract (VO)

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

Résumé

Nous étudions les reparamétrisations $\left({\varphi }_t^{\tau }\right)$ des flots nilpotents de Heisenberg de type borné sur une variété nilpotente de Heisenberg $M$. Nous montrons que, pour des fonctions positives $\tau \in W^s\left(M\right)$ (espace de Sobolev) avec $s>7/2$, toute reparamétrisation non triviale $\left({\varphi }_t^{\tau }\right)$ 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 $\tau \in W^s\left(M\right)$, avec $s>9/2$ et pour tous $p,q\in \mathbb{N}$, $p\ne q$, les flots $\left({\varphi }_{pt}^{\tau }\right)$ et $\left({\varphi }_{qt}^{\tau }\right)$ 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.

Détail

Reçu le : 2019-02-17
Accepté le : 2019-10-09
Publié le : 2019-11-08
DOI : 10.5802/jep.111

Classification : 37C40, 28D10
Keywords: Nilflows, time-changes, Ratner property, multiple mixing, disjointness
Mots-clés : Flots nilpotents, reparamétrisations, propriété de Ratner

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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