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Slope Gap Distribution of Saddle Connections on the 2n-gon

Jonah Berman, Taylor McAdam, Ananth Miller-Murthy, Caglar Uyanik, Hamilton Wan

Abstract

We explicitly compute the limiting slope gap distribution for saddle connections on any 2n-gon. Our calculations show that the slope gap distribution for a translation surface is not always unimodal, answering a question of Athreya. We also give linear lower and upper bounds for number of non-differentiability points as n grows. The latter result exhibits the first example of a non-trivial bound on an infinite family of translation surfaces and answers a question by Kumanduri-Sanchez-Wang.

Slope Gap Distribution of Saddle Connections on the 2n-gon

Abstract

We explicitly compute the limiting slope gap distribution for saddle connections on any 2n-gon. Our calculations show that the slope gap distribution for a translation surface is not always unimodal, answering a question of Athreya. We also give linear lower and upper bounds for number of non-differentiability points as n grows. The latter result exhibits the first example of a non-trivial bound on an infinite family of translation surfaces and answers a question by Kumanduri-Sanchez-Wang.

Paper Structure

This paper contains 15 sections, 13 theorems, 184 equations, 34 figures.

Table of Contents

  1. Introduction
  2. Some History on Slope Gap Distributions
  3. Acknowledgements
  4. Preliminaries
  5. Translation Surfaces and the GL2R action
  6. Flat geometry of 2n-gons and Staircases
  7. From Slope Gaps to Dynamics
  8. Dynamical Reframing
  9. Description of the Poincaré Section
  10. Return time and volume computations
  11. Description of R(x,y) Over Omega2
  12. Description of R(x,y) Over Omega1
  13. Slope Gap Distributions
  14. Bounds on the Number of Non-Differentiability Points
  15. Return Time Function Computations

Key Result

Theorem 1.1

For any regular $2n$-gon $O_{2n}$ where $n\ge3$, there exists a limiting probability distribution function $f: \mathbb{R} \to [0,\infty)$ such that The distribution is a piecewise analytic function with finitely many domains of analyticity that are describable in terms of integrals of elementary functions. Moreover, the distribution has finitely many points of non-differentiability, has no suppor

Figures (34)

  • Figure 1: The regular decagon, $O_{10}$. Black vertices are identified together and red vertices are identified together.
  • Figure 2: A saddle connection $\gamma$ on $X_{10}$
  • Figure 3: An example of $\mathcal{S}$ for $n = 7$.
  • Figure 4: Cylinder decomposition of $O_{2n}$ in the $\theta=\pi/2n$ direction
  • Figure 5: A staircase representative of $O_{12}$
  • ...and 29 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1
  • proof
  • ...and 18 more