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

Siyuan Tang

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arXiv · 2411.03560

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

Siyuan Tang

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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.

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Siyuan Tang. 2024-11-05. Effective density of surfaces near Teichm\"{u}ller curves. https://arxiv.org/abs/2411.03560

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