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Weak mixing directions in non-arithmetic Veech surfaces


Authors: Artur Avila and Vincent Delecroix
Journal: J. Amer. Math. Soc. 29 (2016), 1167-1208
MSC (2010): Primary 37A05, 37E35
DOI: https://doi.org/10.1090/jams/856
Published electronically: January 13, 2016
MathSciNet review: 3522612
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Abstract: We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square, and hexagon). More generally, we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non-arithmetic Veech surface and show that the Hausdorff dimension of the set of non-weak mixing directions is not full. We also provide a necessary condition, verified, for instance, by the Veech surface corresponding to the billiard in the pentagon, for the set of non-weak mixing directions to have a positive Hausdorff dimension.


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Artur Avila
Affiliation: CNRS UMR 7586, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France & IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil
Email: artur@math.jussieu.fr

Vincent Delecroix
Affiliation: CNRS UMR 7586, Instittut de Mathématiques de Jussieu-Paris Rive Gauche, Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
Address at time of publication: LaBRI, UMR 5800, Bãtiment A30, 351, cours de la Libãration 33405 Talence cedex, France.
Email: delecroix@math.jussieu.fr

DOI: https://doi.org/10.1090/jams/856
Keywords: Dynamical systems, eigenvalues, Veech surfaces
Received by editor(s): June 28, 2013
Received by editor(s) in revised form: July 6, 2015, October 4, 2015, and November 30, 2015
Published electronically: January 13, 2016
Additional Notes: The authors were partially supported by the ERC Starting Grant “Quasiperiodic” and by the Balzan project of Jacob Palis.
Article copyright: © Copyright 2016 by the authors under Creative Commons Attribution 3.0 License (CC BY NC ND)