American Institute of Mathematical Sciences

Advanced
  2019, 26: 16-23. doi: 10.3934/era.2019.26.002

Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows

Livio Flaminio 1, and  Giovanni Forni 2,

1. 

Département de Mathématiques, Université de Lille, Cité Scientifique, Villeneuve, D'Ascq, Cedex 9655, FR
2. 

Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, MD 20742-4015, USA

Received  November 06, 2018 Revised  January 25, 2019 Published  March 2019

Fund Project: The first author is partially supported by the Labex CEMPI. The second author is supported by NSF grant DMS 1600687.

We derive, from the work of M. Ratner on joinings of time-changes of horocycle flows and from the result of the authors on its cohomology, the property of orthogonality of powers for non-trivial smooth time-changes of horocycle flows on compact quotients. Such a property is known to imply P. Sarnak's Möbius orthogonality conjecture, already known for horocycle flows by the work of J. Bourgain, P. Sarnak and T. Ziegler.

Keywords: Horocycle flow, cohomological equation, invariant distributions, Möbius orthogonality.
Mathematics Subject Classification: Primary: 11L20, 11N37; Secondary: 37D40.
Citation: Livio Flaminio, Giovanni Forni. Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows. Electronic Research Announcements, 2019, 26: 16-23. doi: 10.3934/era.2019.26.002
References:
[1]

J. Bourgain, P. Sarnak, and T. Ziegler, Disjointness of Moebius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67–83. doi: 10.1007/978-1-4614-4075-8_5.  Google Scholar

[2]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 851-903.  doi: 10.24033/asens.2229.  Google Scholar

[3]

D. Dolgopyat and O. Sarig, Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., 166 (2017), 680-713.  doi: 10.1007/s10955-016-1689-3.  Google Scholar

[4]

E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal., 266 (2014), 284-317.  doi: 10.1016/j.jfa.2013.09.005.  Google Scholar

[5]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[6]

A. Kanigowski, M. Lemańczyk, and C. Ulcigrai, On disjointness properties of some parabolic flows, arXiv: 1810.11576, preprint. Google Scholar

[7]

M. Ratner, Rigid reparametrizations and cohomology for horocycle flows, Invent. Math., 88 (1987), 341-374.  doi: 10.1007/BF01388912.  Google Scholar

[8]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/paper/512, Mathematics - Number Theory, 11N37, 2011. Google Scholar

show all references

References:
[1]

J. Bourgain, P. Sarnak, and T. Ziegler, Disjointness of Moebius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67–83. doi: 10.1007/978-1-4614-4075-8_5.  Google Scholar

[2]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 851-903.  doi: 10.24033/asens.2229.  Google Scholar

[3]

D. Dolgopyat and O. Sarig, Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., 166 (2017), 680-713.  doi: 10.1007/s10955-016-1689-3.  Google Scholar

[4]

E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal., 266 (2014), 284-317.  doi: 10.1016/j.jfa.2013.09.005.  Google Scholar

[5]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[6]

A. Kanigowski, M. Lemańczyk, and C. Ulcigrai, On disjointness properties of some parabolic flows, arXiv: 1810.11576, preprint. Google Scholar

[7]

M. Ratner, Rigid reparametrizations and cohomology for horocycle flows, Invent. Math., 88 (1987), 341-374.  doi: 10.1007/BF01388912.  Google Scholar

[8]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/paper/512, Mathematics - Number Theory, 11N37, 2011. Google Scholar

[1]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[2]

Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158
[3]

Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463
[4]

Konovenko Nadiia, Lychagin Valentin. Möbius invariants in image recognition. Journal of Geometric Mechanics, 2017, 9 (2) : 191-206. doi: 10.3934/jgm.2017008
[5]

Livio Flaminio, Giovanni Forni. On the cohomological equation for nilflows. Journal of Modern Dynamics, 2007, 1 (1) : 37-60. doi: 10.3934/jmd.2007.1.37
[6]

Petr Kůrka. Minimality in iterative systems of Möbius transformations. Conference Publications, 2011, 2011 (Special) : 903-912. doi: 10.3934/proc.2011.2011.903
[7]

Petr Kůrka. Iterative systems of real Möbius transformations. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 567-574. doi: 10.3934/dcds.2009.25.567
[8]

Wen Huang, Jianya Liu, Ke Wang. Möbius disjointness for skew products on a circle and a nilmanifold. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3531-3553. doi: 10.3934/dcds.2021006
[9]

James Tanis, Zhenqi Jenny Wang. Cohomological equation and cocycle rigidity of discrete parabolic actions. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3969-4000. doi: 10.3934/dcds.2019160
[10]

Rich Stankewitz, Hiroki Sumi. Backward iteration algorithms for Julia sets of Möbius semigroups. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6475-6485. doi: 10.3934/dcds.2016079
[11]

Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003
[12]

Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010
[13]

Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021076
[14]

Livio Flaminio, Giovanni Forni, Federico Rodriguez Hertz. Invariant distributions for homogeneous flows and affine transformations. Journal of Modern Dynamics, 2016, 10: 33-79. doi: 10.3934/jmd.2016.10.33
[15]

Slobodan N. Simić. Hölder forms and integrability of invariant distributions. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 669-685. doi: 10.3934/dcds.2009.25.669
[16]

Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123.

[17]

Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4619-4635. doi: 10.3934/dcds.2016001
[18]

Dieter Armbruster, Simone Göttlich, Stephan Knapp. Continuous approximation of $ M_t/M_t/ 1 $ distributions with application to production. Journal of Computational Dynamics, 2020, 7 (2) : 243-269. doi: 10.3934/jcd.2020010
[19]

Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Corrigendum to "Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology". Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4585-4586. doi: 10.3934/dcds.2017196
[20]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483

2019 Impact Factor: 0.5

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences