Effective Exponential Drifts on Strata of Abelian Differentials

TL;DR

This paper investigates the dynamics of the $SL_{2}( ext{R})$ action on the stratum $ ext{H}(2)$ of translation surfaces, demonstrating effective exponential drift and discretized orbit dimensions using advanced homogeneous dynamics techniques.

Contribution

It provides an effective version of exponential drift on $ ext{H}(2)$ and establishes discretized orbit dimensions via an effective closing lemma and equidistribution theorems.

Findings

Effective exponential drift on $ ext{H}(2)$ established.

Discretized orbit dimension of almost 1 in a transverse direction.

Utilizes McMullen's classification and effective equidistribution results.

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.