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Super-fast spreading of billiard orbits in rational polygons and geodesics on translation surfaces

J. Beck, W. W. L. Chen

Abstract

In this paper, we show that billiard orbits in rational polygons and geodesics on translation surfaces exhibit super-fast spreading, an optimal time-quantitative majority property about the corresponding linear flow that implies uniformity in almost every direction.

Super-fast spreading of billiard orbits in rational polygons and geodesics on translation surfaces

Abstract

In this paper, we show that billiard orbits in rational polygons and geodesics on translation surfaces exhibit super-fast spreading, an optimal time-quantitative majority property about the corresponding linear flow that implies uniformity in almost every direction.
Paper Structure (7 sections, 13 theorems, 267 equations)

This paper contains 7 sections, 13 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Geodesics on a translation surface
  3. Partitions, balance and anti-crowdedness
  4. Starting the proof of super-fast spreading
  5. Studying Case 1
  6. Studying Case 2
  7. Completing the proof

Key Result

Theorem 1

Let $\mathcal{P}$ be an arbitrary translation surface of area $1$. Then geodesic flow on $\mathcal{P}$ exhibits super-fast spreading.

Theorems & Definitions (23)

  • Theorem 1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 13 more