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Effective density of surfaces near Teichmüller curves

Siyuan Tang

Abstract

We study the dynamics of $SL_{2}(\mathbb{R})$ on the stratum of translation surfaces $\mathcal{H}(2)$. Especially, we obtain effective density theorems on $\mathcal{H}(2)$ for orbits of the upper triangular subgroup $P$ of $SL_{2}(\mathbb{R})$ with the based surfaces near a small Teichmüller curve. The proof is based on the use of McMullen's classification theorem, together with the effective equidistribution theorems in homogeneous dynamics. In particular, we compare the $P$-orbit of a surface, and the $P$-orbit of its absolute periods using the Lindenstrauss-Mohammadi-Wang's effective equidistribution theorem.

Effective density of surfaces near Teichmüller curves

Abstract

We study the dynamics of on the stratum of translation surfaces . Especially, we obtain effective density theorems on for orbits of the upper triangular subgroup of with the based surfaces near a small Teichmüller curve. The proof is based on the use of McMullen's classification theorem, together with the effective equidistribution theorems in homogeneous dynamics. In particular, we compare the -orbit of a surface, and the -orbit of its absolute periods using the Lindenstrauss-Mohammadi-Wang's effective equidistribution theorem.

Paper Structure

This paper contains 28 sections, 59 theorems, 338 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof of Theorem \ref{['effective2024.9.1']}
  4. Outline of the proof of Theorem \ref{['effective2024.07.148']}
  5. Structure of the paper
  6. Preliminaries
  7. Notation
  8. Translation surfaces
  9. Teichmüller curves
  10. Nondivergence
  11. Avila-Gouëzel-Yoccoz norm
  12. Triangulation
  13. Dynamics over $\mathcal{H}(2)$
  14. McMullen's classification
  15. Prototypes of eigenforms
  16. ...and 13 more sections

Key Result

Theorem 1.1

For any $\eta>0$, $L>0$, there exists $\kappa_{1}>0$, $C_{1}=C_{1}(\eta)>0$ such that for $D\geq L^{2024.08.k1}$, and the following holds: Suppose that $x\in\mathcal{H}_{1}(2)$ satisfying Then we have for every $z\in \mathcal{H}_{1}^{(L^{-1})}(2)$.

Figures (6)

  • Figure 1: Outline of the proof of Theorem \ref{['effective2024.9.1']}.
  • Figure 2: Quantitative nondivergence of horocycle flows on $\mathcal{H}_{1}(2)$.
  • Figure 3: A Delaunay triangulation of a surface in $\mathcal{H}(2)$.
  • Figure 4: Prototypical splitting of type $(a,b,c,e)$.
  • Figure 5: Pair of splittings.
  • ...and 1 more figures

Theorems & Definitions (103)

  • Theorem 1.1
  • Conjecture 1.2: McMullen's expansion conjecture
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Translation surface
  • Definition 2.2: Teichmüller curve
  • Theorem 2.3
  • Theorem 2.4: lindenstrauss2023polynomial
  • Theorem 2.5: eskin2001asymptoticathreya2006quantitative
  • ...and 93 more