Effective Exponential Drifts on Strata of Abelian Differentials

We study the dynamics of $S{L}_2(\mathbb{R})$ on the stratum of translation surfaces $\mathcal{H}(2)$. In particular, we prove that an orbit of the upper triangular subgroup of $S{L}_2(\mathbb{R})$ has a discretized dimension of almost $1$ in a direction transverse to the $S{L}_2(\mathbb{R})$-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.