Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space

Abstract

In this paper we compute the Hausdorff dimension of the set $D(\varphi_n)$ of points on divergent trajectories of the homogeneous flow $\varphi_n$ induced by the one-parameter subgroup $\mathop{\rm diag}(e^t,e^{-t})$ acting by left multiplication on the product space $G^n/\Gamma^n$, where $G=\mathop{\rm SL}(2,{\mathbb R})$ and $\Gamma=\mathop{\rm SL}(2,{\mathbb Z})$. We prove that $\dim_H D(\varphi_n)=3n-\frac{1}{2}$ for $n\ge2$.