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

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 τW 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 τW s (M), avec s>9/2 et pour tous p,q, pq, les flots (ϕ pt τ ) et (ϕ qt 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 τW 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 τW s (M), s>9/2 and every p,q, pq, (ϕ pt τ ) and (ϕ qt τ ) are disjoint. As a consequence, Sarnak conjecture on Möbius disjointness holds for all such time changes.

Reçu le :
Accepté le :
Publié le :
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
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
@article{JEP_2020__7__63_0,
     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/}
}
TY  - JOUR
AU  - Giovanni Forni
AU  - Adam Kanigowski
TI  - Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2020
SP  - 63
EP  - 91
VL  - 7
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.111/
DO  - 10.5802/jep.111
LA  - en
ID  - JEP_2020__7__63_0
ER  - 
%0 Journal Article
%A Giovanni Forni
%A Adam Kanigowski
%T Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows
%J Journal de l’École polytechnique — Mathématiques
%D 2020
%P 63-91
%V 7
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.111/
%R 10.5802/jep.111
%G en
%F JEP_2020__7__63_0
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/

[1] H. Anzai - “Ergodic skew product transformations on the torus”, Osaka J. Math. 3 (1951), p. 83-99 | MR | Zbl

[2] L. Auslander, L. Green & F. Hahn - Flows on homogeneous spaces, Annals of Math. Studies, vol. 53, Princeton University Press, Princeton, NJ, 1963 | MR | Zbl

[3] A. Avila, G. Forni & C. Ulcigrai - “Mixing for time-changes of Heisenberg nilflows”, J. Differential Geom. 89 (2011) no. 3, p. 369-410 | DOI | MR | Zbl

[4] A. Avila, D. Ravotti, G. Forni & C. Ulcigrai - “Mixing for smooth time-changes of general nilflows”, 2019 | arXiv

[5] J. Bourgain, P. Sarnak & T. Ziegler - “Disjointness of Moebius from horocycle flows”, in From Fourier analysis and number theory to Radon transforms and geometry, Dev. Math., vol. 28, Springer, New York, 2013, p. 67-83 | DOI | MR | Zbl

[6] B. Fayad, G. Forni & A. Kanigowski - “Lebesgue spectrum of countable multiplicity for conservative flows on the torus”, 2016 | arXiv

[7] B. Fayad & A. Kanigowski - “Multiple mixing for a class of conservative surface flows”, Invent. Math. 203 (2016) no. 2, p. 555-614 | DOI | MR | Zbl

[8] S. Ferenczi, J. Kułaga-Przymus & M. Lemańczyk - “Sarnak’s conjecture: what’s new”, in Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics (CIRM Jean-Morlet Chair, Fall 2016), Lect. Notes in Math., vol. 2213, Springer, Cham, 2018, p. 163-235 | DOI | MR | Zbl

[9] L. Flaminio & G. Forni - “Equidistribution of nilflows and applications to theta sums”, Ergodic Theory Dynam. Systems 26 (2006) no. 2, p. 409-433 | DOI | MR | Zbl

[10] L. Flaminio & G. Forni - “Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows”, Electron. Res. Announc. Math. Sci. 26 (2019), p. 16-23 | Zbl

[11] L. Flaminio, K. Frączek, J. Kułaga-Przymus & M. Lemańczyk - “Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds”, Studia Math. 244 (2019) no. 1, p. 43-97 | DOI | MR | Zbl

[12] G. Forni - “Effective equidistribution of nilflows and bounds on Weyl sums”, in Dynamics and analytic number theory, London Math. Soc. Lecture Note Ser., vol. 437, Cambridge Univ. Press, Cambridge, 2016, p. 136-188 | DOI | MR | Zbl

[13] G. Forni & A. Kanigowski - “Time-changes of Heisenberg nilflows”, 2017 | arXiv

[14] G. Forni & C. Ulcigrai - “Time-changes of horocycle flows”, J. Modern Dyn. 6 (2012) no. 2, p. 251-273 | DOI | MR | Zbl

[15] P. Gabriel, M. Lemańczyk & K. Schmidt - “Extensions of cocycles for hyperfinite actions and applications”, Monatsh. Math. 123 (1997) no. 3, p. 209-228 | DOI | MR | Zbl

[16] E. Glasner - Ergodic theory via joinings, Math. Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003 | DOI | MR | Zbl

[17] B. Green & T. Tao - “The Möbius function is strongly orthogonal to nilsequences”, Ann. of Math. (2) 175 (2012) no. 2, p. 541-566 | DOI | Zbl

[18] B. Hasselblatt & A. Katok - “Principal structures”, in Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, p. 1-203 | DOI | Zbl

[19] A. Kanigowski & J. Kułaga-Przymus - “Ratner’s property and mild mixing for smooth flows on surfaces”, Ergodic Theory Dynam. Systems 36 (2016) no. 8, p. 2512-2537 | DOI | MR | Zbl

[20] A. Kanigowski, J. Kułaga-Przymus & C. Ulcigrai - “Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces”, J. Eur. Math. Soc. (JEMS) 21 (2019) no. 12, p. 3797-3855 | DOI | MR | Zbl

[21] A. Kanigowski & M. Lemańczyk - “Flows with Ratner’s property have discrete essential centralizer”, Studia Math. 237 (2017) no. 2, p. 185-194 | DOI | MR | Zbl

[22] A. Kanigowski, M. Lemańczyk & C. Ulcigrai - “On disjointness properties of some parabolic flows”, 2018 | arXiv

[23] I. Kátai - “A remark on a theorem of H. Daboussi”, Acta Math. Hungar. 47 (1986) no. 1-2, p. 223-225 | DOI | MR | Zbl

[24] A. Katok - Combinatorial constructions in ergodic theory and dynamics, University Lect. Series, vol. 30, American Mathematical Society, Providence, RI, 2003 | DOI | MR | Zbl

[25] M. Ratner - “Horocycle flows, joinings and rigidity of products”, Ann. of Math. (2) 118 (1983) no. 2, p. 277-313 | DOI | MR | Zbl

[26] M. Ratner - “Rigid reparametrizations and cohomology for horocycle flows”, Invent. Math. 88 (1987) no. 2, p. 341-374 | DOI | MR | Zbl

[27] D. Ravotti - “Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows”, Ergodic Theory Dynam. Systems 39 (2018) no. 12, p. 3407-3436 | DOI | MR | Zbl

[28] P. Sarnak - “Three lectures on the Möbius function randomness and dynamics”, 2011, https://publications.ias.edu/sarnak/paper/512

[29] R. Tiedra de Aldecoa - “Spectral analysis of time changes of horocycle flows”, J. Modern Dyn. 6 (2012) no. 2, p. 275-285 | DOI | MR | Zbl

[30] R. Tiedra de Aldecoa - “Commutator methods for the spectral analysis of uniquely ergodic dynamical systems”, Ergodic Theory Dynam. Systems 35 (2015) no. 3, p. 944-967 | DOI | MR | Zbl

Cité par Sources :