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Weak mixing in rational billiards

Francisco Arana-Herrera, Jon Chaika, Giovanni Forni

Abstract

We completely characterize rational polygons whose billiard flow is weakly mixing in almost every direction as those which are not almost integrable, in the terminology of Gutkin, modulo some low complexity exceptions. This proves a longstanding conjecture of Gutkin. This result is derived from a complete characterization of translation surfaces that are weakly mixing in almost every direction: they are those that do not admit an affine factor map to the circle.

Weak mixing in rational billiards

Abstract

We completely characterize rational polygons whose billiard flow is weakly mixing in almost every direction as those which are not almost integrable, in the terminology of Gutkin, modulo some low complexity exceptions. This proves a longstanding conjecture of Gutkin. This result is derived from a complete characterization of translation surfaces that are weakly mixing in almost every direction: they are those that do not admit an affine factor map to the circle.

Paper Structure

This paper contains 22 sections, 26 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Statement of results
  4. Related results
  5. Sketch of proof
  6. Acknowledgments
  7. Statement of the main theorem and reductions
  8. Outline of this section.
  9. Background
  10. Statement of the main theorem
  11. Criterion for weak mixing
  12. Reduction to positive Lyapunov exponents
  13. From rotations to horocycles
  14. Proof of (2) implies (1) in Theorem \ref{['thm:main1']}
  15. The existence of $(V,H,\sigma,L)$-rigidity configurations
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $(X,\omega)$ be a translation surface and $F_\theta$ denote its flow in the direction $\theta \in [0,2\pi)$. Then, the following are equivalent:

Figures (1)

  • Figure 1: Rigidity configuration on a translation surface.

Theorems & Definitions (57)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 3.1
  • proof
  • proof
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 47 more