We study time changes of bounded type Heisenberg nilflows acting on the Heisenberg nilmanifold . We show that for every positive , , every non-trivial time change enjoys the Ratner property. As a consequence, every mixing time change is mixing of all orders. Moreover, we show that for every , and every , , and are disjoint. As a consequence, Sarnak conjecture on Möbius disjointness holds for all such time changes.
Nous étudions les reparamétrisations des flots nilpotents de Heisenberg de type borné sur une variété nilpotente de Heisenberg . Nous montrons que, pour des fonctions positives (espace de Sobolev) avec , toute reparamétrisation non triviale 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 , avec et pour tous , , les flots et 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.
@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 TI - Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows JO - Journal de l’École polytechnique — Mathématiques PY - 2020 DA - 2020/// SP - 63 EP - 91 VL - 7 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.111/ UR - https://zbmath.org/?q=an%3A07128377 UR - https://www.ams.org/mathscinet-getitem?mr=4033750 UR - https://doi.org/10.5802/jep.111 DO - 10.5802/jep.111 LA - en ID - JEP_2020__7__63_0 ER -
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/
[1] - “Ergodic skew product transformations on the torus”, Osaka J. Math. 3 (1951), p. 83-99 | MR: 40594 | Zbl: 0043.11203
[2] - Flows on homogeneous spaces, Annals of Math. Studies, vol. 53, Princeton University Press, Princeton, NJ, 1963 | MR: 167569 | Zbl: 0099.39103
[3] - “Mixing for time-changes of Heisenberg nilflows”, J. Differential Geom. 89 (2011) no. 3, p. 369-410 | Article | MR: 2879246 | Zbl: 1281.37012
[4] - “Mixing for smooth time-changes of general nilflows”, 2019 | 1905.11628
[5] - “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 | Article | MR: 2986954 | Zbl: 1336.37030
[6] - “Lebesgue spectrum of countable multiplicity for conservative flows on the torus”, 2016 | 1609.03757v2
[7] - “Multiple mixing for a class of conservative surface flows”, Invent. Math. 203 (2016) no. 2, p. 555-614 | Article | MR: 3455157 | Zbl: 1358.37018
[8] - “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 | Article | MR: 3821717 | Zbl: 1421.37003
[9] - “Equidistribution of nilflows and applications to theta sums”, Ergodic Theory Dynam. Systems 26 (2006) no. 2, p. 409-433 | Article | MR: 2218767 | Zbl: 1087.37026
[10] - “Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows”, Electron. Res. Announc. Math. Sci. 26 (2019), p. 16-23 | Zbl: 07138890
[11] - “Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds”, Studia Math. 244 (2019) no. 1, p. 43-97 | Article | MR: 3850672 | Zbl: 06970467
[12] - “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 | Article | MR: 3618789 | Zbl: 06984497
[13] - “Time-changes of Heisenberg nilflows”, 2017 | 1711.05543v1
[14] - “Time-changes of horocycle flows”, J. Modern Dyn. 6 (2012) no. 2, p. 251-273 | Article | MR: 2968956 | Zbl: 1254.37006
[15] - “Extensions of cocycles for hyperfinite actions and applications”, Monatsh. Math. 123 (1997) no. 3, p. 209-228 | Article | MR: 1442000 | Zbl: 0887.28008
[16] - Ergodic theory via joinings, Math. Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003 | Article | MR: 1958753 | Zbl: 1038.37002
[17] - “The Möbius function is strongly orthogonal to nilsequences”, Ann. of Math. (2) 175 (2012) no. 2, p. 541-566 | Article | Zbl: 1347.37019
[18] - “Principal structures”, in Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, p. 1-203 | Article | Zbl: 1047.37001
[19] - “Ratner’s property and mild mixing for smooth flows on surfaces”, Ergodic Theory Dynam. Systems 36 (2016) no. 8, p. 2512-2537 | Article | MR: 3570023 | Zbl: 1362.37011
[20] - “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 | Article | MR: 4022716 | Zbl: 1427.37035
[21] - “Flows with Ratner’s property have discrete essential centralizer”, Studia Math. 237 (2017) no. 2, p. 185-194 | Article | MR: 3620751 | Zbl: 1375.37005
[22] - “On disjointness properties of some parabolic flows”, 2018 | 1810.11576
[23] - “A remark on a theorem of H. Daboussi”, Acta Math. Hungar. 47 (1986) no. 1-2, p. 223-225 | Article | MR: 836415 | Zbl: 0607.10034
[24] - Combinatorial constructions in ergodic theory and dynamics, University Lect. Series, vol. 30, American Mathematical Society, Providence, RI, 2003 | Article | MR: 2008435 | Zbl: 1030.37001
[25] - “Horocycle flows, joinings and rigidity of products”, Ann. of Math. (2) 118 (1983) no. 2, p. 277-313 | Article | MR: 717825 | Zbl: 0556.28020
[26] - “Rigid reparametrizations and cohomology for horocycle flows”, Invent. Math. 88 (1987) no. 2, p. 341-374 | Article | MR: 880955 | Zbl: 0697.58046
[27] - “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 | Article | MR: 4027556 | Zbl: 07128556
[28] - “Three lectures on the Möbius function randomness and dynamics”, 2011, https://publications.ias.edu/sarnak/paper/512
[29] - “Spectral analysis of time changes of horocycle flows”, J. Modern Dyn. 6 (2012) no. 2, p. 275-285 | Article | MR: 2968957 | Zbl: 1290.37013
[30] - “Commutator methods for the spectral analysis of uniquely ergodic dynamical systems”, Ergodic Theory Dynam. Systems 35 (2015) no. 3, p. 944-967 | Article | MR: 3334911 | Zbl: 1355.37014
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