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

Forni, Giovanni; Kanigowski, Adam

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

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.

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.

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

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