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Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit

David Aulicino, Howard Masur, Huiping Pan, Weixu Su

Abstract

We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an $L (\log L)^{d-2}$ growth rate, where $d$ is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization.

Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit

Abstract

We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an growth rate, where is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization.
Paper Structure (48 sections, 42 theorems, 155 equations, 3 figures)

This paper contains 48 sections, 42 theorems, 155 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Heuristic
  3. Translation Surfaces and Strata
  4. Translation Surfaces
  5. Moduli Space of Translation Surfaces
  6. Hyperelliptic Translation Surfaces
  7. Definitions
  8. Results
  9. Notation and Conventions
  10. Auxiliary Sets of Saddle Connections
  11. Partitioning $\Gamma$ -- $\epsilon$-isolated:
  12. Partitioning $\Gamma$ -- $\epsilon$-non-isolated:
  13. Forming $\Sigma_k$ and its Partition:
  14. Termination of the Construction:
  15. Upper Bound Theorems
  16. ...and 33 more sections

Key Result

Theorem 1.1

Suppose $(X, \omega) \in \mathcal{H}^{hyp}_1(\kappa)$ and $\beta$ is a marked saddle connection on $(X, \omega)$ satisfying $\tau(\beta) = \beta$. Let $d = \dim_\mathbb{C} \mathcal{H}^{hyp}(\kappa)$. Let $A((X, \omega) \setminus \beta,L)$ be the set of saddle connections interiorly disjoint from $\b

Figures (3)

  • Figure 1: A decomposition of a translation surface in $\mathcal{H}^{hyp}(4)$ into five parallelograms cf. Lemma \ref{['Parallelogramulation:Lemma']}.
  • Figure 2: Relative positions of triangles determined by $\gamma$ and $\gamma'$.
  • Figure 3: Case (a) with $\gamma'$ sheared to a vertical saddle connection.

Theorems & Definitions (104)

  • Theorem 1.1
  • Remark
  • Remark
  • Definition
  • Example 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition
  • Definition
  • Remark
  • ...and 94 more