Weak mixing directions in non-arithmetic Veech surfaces

Avila, Artur¹ ; Delecroix, Vincent^2, 3

¹ 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
² CNRS UMR 7586, Instittut de Mathématiques de Jussieu-Paris Rive Gauche, Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
³ LaBRI, UMR 5800, Bãtiment A30, 351, cours de la Libãration 33405 Talence cedex, France.

Journal of the American Mathematical Society, Tome 29 (2016) no. 4, pp. 1167-1208

Résumé

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.

Détail

DOI : 10.1090/jams/856

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