# American Institute of Mathematical Sciences

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

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

 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.

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
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