Effective density of surfaces near Teichm\"{u}ller curves

TL;DR

This paper investigates the distribution of surfaces near Teichmüller curves within the stratum (2), providing effective density results for certain group orbits using advanced homogeneous dynamics techniques.

Contribution

It establishes effective density theorems for $P$-orbits near Teichmüller curves in (2) by combining McMullen's classification with effective equidistribution theorems.

Findings

Effective density theorems for $P$-orbits near Teichmüller curves

Comparison of $P$-orbits of surfaces and their absolute periods

Application of homogeneous dynamics and equidistribution results

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 Teichm\"{u}ller 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.