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Effective Exponential Drifts on Strata of Abelian Differentials

Siyuan Tang

TL;DR

This work analyzes the $SL_2(\\mathbb{R})$-action on the genus-2 translation-surface stratum $\\mathcal{H}(2)$, establishing an effective exponential-drift phenomenon for upper-triangular dynamics. By combining an effective closing lemma with Margulis-function techniques and leveraging McMullen’s classification alongside Lindenstrauss–Mohammadi–Wang effective equidistribution, the authors show that the discretized transversal dimension along the $P=a_{\\mathbb{R}}u_{\\mathbb{R}}$-orbit is nearly 1; in the genus-2 case, they further deduce the existence of nearby Teichmüller curves with controlled discriminants under certain conditions. The paper develops a comprehensive toolkit—AGY norm, spectral-gap based lattice-point counts, nondivergence estimates, and a Margulis-function framework—to iteratively enhance the dimension of invariance, ultimately connecting dynamics on $\\mathcal{H}_1(2)$ to Teichmüller curves and their arithmetic data. This yields quantitative discreteness results for Teichmüller curves, effective closing lemmas, and density-type statements for unstable foliations, with explicit implications for discriminant bounds and the structure of Weierstrass curves. The results advance the understanding of transverse dynamics in moduli spaces of flat surfaces and provide effective mechanisms to link dynamics with arithmetic geometry via discriminants and Veech groups.

Abstract

We study the dynamics of $SL_{2}(\mathbb{R})$ on the stratum of translation surfaces $\mathcal{H}(2)$. In particular, we prove that an orbit of the upper triangular subgroup of $SL_{2}(\mathbb{R})$ has a discretized dimension of almost $1$ in a direction transverse to the $SL_{2}(\mathbb{R})$-orbit. The proof proceeds via an effective closing lemma, and the Margulis function technique, which serves as an effective version of the exponential drift on $\mathcal{H}(2)$. The idea is based on the use of McMullen's classification theorem, together with Lindenstrauss-Mohammadi-Wang's effective equidistribution theorems in homogeneous dynamics.

Effective Exponential Drifts on Strata of Abelian Differentials

TL;DR

This work analyzes the -action on the genus-2 translation-surface stratum , establishing an effective exponential-drift phenomenon for upper-triangular dynamics. By combining an effective closing lemma with Margulis-function techniques and leveraging McMullen’s classification alongside Lindenstrauss–Mohammadi–Wang effective equidistribution, the authors show that the discretized transversal dimension along the -orbit is nearly 1; in the genus-2 case, they further deduce the existence of nearby Teichmüller curves with controlled discriminants under certain conditions. The paper develops a comprehensive toolkit—AGY norm, spectral-gap based lattice-point counts, nondivergence estimates, and a Margulis-function framework—to iteratively enhance the dimension of invariance, ultimately connecting dynamics on to Teichmüller curves and their arithmetic data. This yields quantitative discreteness results for Teichmüller curves, effective closing lemmas, and density-type statements for unstable foliations, with explicit implications for discriminant bounds and the structure of Weierstrass curves. The results advance the understanding of transverse dynamics in moduli spaces of flat surfaces and provide effective mechanisms to link dynamics with arithmetic geometry via discriminants and Veech groups.

Abstract

We study the dynamics of on the stratum of translation surfaces . In particular, we prove that an orbit of the upper triangular subgroup of has a discretized dimension of almost in a direction transverse to the -orbit. The proof proceeds via an effective closing lemma, and the Margulis function technique, which serves as an effective version of the exponential drift on . The idea is based on the use of McMullen's classification theorem, together with Lindenstrauss-Mohammadi-Wang's effective equidistribution theorems in homogeneous dynamics.
Paper Structure (29 sections, 59 theorems, 382 equations, 1 figure)

This paper contains 29 sections, 59 theorems, 382 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of Theorem \ref{['margulis2024.12.1']}
  4. Structure of the paper
  5. Preliminaries
  6. Notation
  7. Translation surfaces
  8. Dynamics of Teichmüller flow
  9. Teichmüller curves
  10. Nondivergence
  11. Avila-Gouëzel-Yoccoz norm
  12. Triangulation
  13. Spectral gap and effective lattice point counting
  14. Basic properties of spectral gap
  15. Effective lattice point counting
  16. ...and 14 more sections

Key Result

Theorem 1.1

For $\alpha=(2g-2)$, there exist $N_{0}=N_{0}(\alpha)$ that satisfy the following. For any $\epsilon\in(0,\frac{1}{10})$, $\gamma\in(0,1)$, $\eta\in(0, N_{0}^{-1})$, $N\geq N_{0}$, $x_{0}\in \mathcal{H}_{1}(\alpha)$, there exist $\kappa_{1}=\kappa_{1}(N,\gamma,\alpha,\epsilon)>0$, $\varkappa =\vark

Figures (1)

  • Figure 2: Prototypical splitting of type $(a,b,c,e)$.

Theorems & Definitions (118)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Definition 2.2: Translation surface
  • Theorem 2.3: Semisimplicity of $\mathop{\mathrm{SL}}\nolimits_{2}(\mathbb{R})$-invariant subbundles, filip2016semisimplicity
  • Theorem 2.4: forni2002deviation
  • Definition 2.5: Teichmüller curve
  • Theorem 2.6
  • ...and 108 more