[Paper Review] Weak mixing directions in non-arithmetic Veech surfaces

Artur Avila, Vincent Delecroix|arXiv (Cornell University)|Apr 11, 2013

Mathematical Dynamics and Fractals21 references4 citations

TL;DR

This paper establishes that billiard flows in regular polygons with $ n \neq 3,4,6 $ sides are weak mixing on almost every invariant surface, resolving a long-standing question in dynamical systems. Using advanced techniques from Teichmüller theory and the Kontsevich-Zorich cocycle, the authors show that the set of non-weak mixing directions has Hausdorff dimension strictly less than one, even when positive, and provide a necessary condition for such directions to have positive dimension, verified for the pentagon surface.

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.

Motivation & Objective

• To resolve the long-standing question of whether billiard flows in regular polygons (except triangle, square, hexagon) are weak mixing on most invariant surfaces.
• To understand the prevalence and geometric structure of non-weak mixing directions in non-arithmetic Veech surfaces.
• To establish a necessary condition for the set of non-weak mixing directions to have positive Hausdorff dimension, applicable to surfaces like the pentagon.
• To extend results on weak mixing beyond generic settings to highly symmetric, rigid systems such as regular polygon billiards.

Proposed method

• Analyzes directional flows on translation surfaces via the $SL(2,\mathbb{R})$-action on the moduli space and the associated Kontsevich-Zorich cocycle.
• Applies a Markov model for the geodesic flow on $\mathrm{SL}(2,\mathbb{R})/\Gamma$ to control recurrence and large deviations in the cocycle behavior.
• Uses a dual Veech criterion involving tunneling curves and eigenfunctions to detect non-weak mixing directions.
• Employs a dynamical approach to bound the Hausdorff dimension of exceptional sets via large deviation estimates and anomalous Lyapunov behavior.
• Constructs explicit examples of non-weak mixing directions using Salem elements in triangle groups, verified via computational algebra in Sage.
• Applies results on holonomy fields and conjugates of Veech groups to characterize the algebraic structure of eigenvalues in exceptional directions.

Experimental results

Research questions

• RQ1 Is the billiard flow weak mixing on almost every invariant surface for regular polygons with $ n \neq 3,4,6 $?
• RQ2 What is the Hausdorff dimension of the set of non-weak mixing directions in non-arithmetic Veech surfaces?
• RQ3 Can a necessary algebraic condition be identified for the set of non-weak mixing directions to have positive Hausdorff dimension?
• RQ4 Does the pentagon Veech surface admit non-weak mixing directions, and if so, what is the structure of this set?
• RQ5 How does the interplay between spectral theory, Teichmüller dynamics, and Diophantine properties constrain the existence of eigenfunctions in exceptional directions?

Key findings

• For $ n \neq 3,4,6 $, the billiard flow in a regular $ n $-gon is weak mixing on almost every invariant surface, resolving a key open problem.
• The set of non-weak mixing directions in any non-arithmetic Veech surface has Hausdorff dimension strictly less than one, even when positive.
• A necessary condition for the set of non-weak mixing directions to have positive Hausdorff dimension is identified, involving the existence of Salem elements in the Veech group.
• The condition is verified for the Veech surface associated with the regular pentagon, confirming that its non-weak mixing directions can have positive dimension.
• Explicit examples of non-weak mixing directions are constructed using Salem elements in triangle groups $ \Delta(2,q,\infty) $ and $ \Delta(q,\infty,\infty) $, with minimal polynomials computed via computational algebra.
• The paper provides a complete classification of such Salem elements up to $ q=17 $ for $ \Delta(2,q,\infty) $ and $ q=16 $ for $ \Delta(q,\infty,\infty) $, using Sage computations.