Bounded and Divergent Trajectories And Expanding Curves on Homogeneous Spaces

Osama Khalil

Published 2018-06-18Version 1

Suppose $g_t$ is a $1$-parameter $\mathrm{Ad}$-diagonalizable subgroup of a Lie group $G$ and $\Gamma < G$ is a lattice. We study differentiable curves of the form $\varphi: [0,1] \rightarrow U^+$ satisfying certain non-degeneracy conditions, where $U^+$ is the expanding horospherical subgroup of $g_t$. For a class of examples that includes products of real rank one Lie groups, we show that for any basepoint $x_0\in G/\Gamma$, the Hausdorff dimension of the set of points $s$ for which the forward orbit $g_t \varphi(s)x_0$ is divergent on average in $G/\Gamma$ is at most $1/2$. Moreover, we prove that the set of points $s$ for which $g_t \varphi(s)x_0$ remains bounded is winning in the sense of Schmidt. We describe applications of our results to problems in diophantine approximation. Our methods also yield the following result for square systems of linear forms: suppose $\varphi(s) = sY + Z$ where $Y\in \mathrm{GL}(n,\mathbb{R})$ and $Z\in M_{n,n}(\mathbb{R})$. Then, the dimension of the set of points $s$ such that $\varphi(s)$ is singular is at most $1/2$ while badly approximable points have Hausdorff dimension equal to $1$.