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 Zbl

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

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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

References
Cited by

[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

Cited by Sources: