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

Giovanni Forni; Adam Kanigowski

doi:10.5802/jep.111

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]

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

Résumé
Abstract

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.

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.

Reçu le : 2019-02-17
Accepté le : 2019-10-09
Publié le : 2019-11-08

MR Zbl

DOI : 10.5802/jep.111

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

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Giovanni Forni and Adam Kanigowski},
     title = {Multiple mixing and disjointness for time changes of bounded-type {Heisenberg} nilflows},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {63--91},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.111},
     zbl = {07128377},
     mrnumber = {4033750},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.111/}
}

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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, Tome 7 (2020), pp. 63-91. doi : 10.5802/jep.111. https://jep.centre-mersenne.org/articles/10.5802/jep.111/

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