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Effective equidistribution of twisted horocycle flows and horocycle maps

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Abstract

We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of Venkatesh, recently already improved by Tanis and Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by Shah and Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cusp forms which are comparable to the best known bounds of Good in the holomorphic case, and of Bernstein and Reznikov in the Maass (non-holomorphic) case. Our approach is based on Sobolev estimates for solutions of the cohomological equation and on scaling of invariant distributions for twisted horocycle flows.

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Authors and Affiliations

  1. Unité Mixte de Recherche CNRS 8524, Unité de Formation et Recherche de Mathématiques, Université de Lille 1, 59655, Villeneuve d’Asq Cedex, France

    Livio Flaminio

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Giovanni Forni

  3. Collège de France, 3 Rue d’Ulm, 75005, Paris, France

    James Tanis

Authors
  1. Livio Flaminio
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  2. Giovanni Forni
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  3. James Tanis
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Correspondence to Giovanni Forni.

Additional information

L. Flaminio was supported in part by the Labex CEMPI. G. Forni was supported by the NSF grant DMS 1201534 and by a Simons Fellowship. J. Tanis was partially supported by the ANR grant ’GeoDyM’ (ANR-11-BS01-0004).

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Flaminio, L., Forni, G. & Tanis, J. Effective equidistribution of twisted horocycle flows and horocycle maps. Geom. Funct. Anal. 26, 1359–1448 (2016). https://doi.org/10.1007/s00039-016-0385-4

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  • DOI: https://doi.org/10.1007/s00039-016-0385-4

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