Weak mixing directions in non-arithmetic Veech surfaces

Artur Avila,Vincent Delecroix

doi:10.1090/jams/856

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.

Highlights

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
In this paper we address directly the problem of weak mixing for exceptionally symmetric translation flows, which include the ones arising from regular polygonal billiards
For a non-arithmetic Veech surface and along any minimal direction that is not weak mixing, there are always exactly [k : Q] independent eigenvalues, and that they are either all continuous or all discontinuous. This is the case along directions for which the corresponding forward Teichmuller geodesic is bounded in moduli space, see Remark 7.1

Summary

Introduction

The set of directions for which the directional flow is not even topologically weak mixing has positive Hausdorff dimension Notice that this covers the case of certain polygonal billiards For a non-arithmetic Veech surface and along any minimal direction that is not weak mixing, there are always exactly [k : Q] independent eigenvalues, and that they are either all continuous or all discontinuous. This is the case along directions for which the corresponding forward Teichmuller geodesic is bounded in moduli space, see Remark 7.1. See [BDM2], section 6, for a different example in a related context

Preliminaries

Eigenfunctions in Veech surfaces

Generic weak-mixing

Markov model

Construction of directions with non-trivial eigenfunctions