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Weak mixing directions in non-arithmetic Veech surfaces

By Artur Avila and Vincent Delecroix

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 positive Hausdorff dimension

Topics: Mathematics - Dynamical Systems, 37A05, 37E35
Year: 2015
OAI identifier: oai:arXiv.org:1304.3318

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