Vasiliy Nekrasov
Brandeis University
Title
Bounded trajectories on homogeneous spaces for subsets without semigroup structure
Abstract
Let $G$ be a connected semisimple real Lie group, $\Gamma$ an irreducible lattice in $G$ and $X = G/\Gamma$ be the corresponding homogeneous space. Let $F = \{g_t: t\ge 0\}$ be a non-quasiunipotent one-parameter subsemigroup of $G$. Kleinbock and Margulis proved that the set of points in $X$ with bounded $F$-trajectories is thick (has full Hausdorff dimension in every neighborhood). In addition, if $U$ is the expanding horospherical subgroup relative to $g_1$, then for any $x \in X$ the set of points $u \in U$ such that the $F$-trajectory of $ux$ is bounded has full Hausdorff dimension. In this talk we raise a natural question: what happens if the set $F$ is not assumed to be a semigroup? It turns out that certain geometric restrictions on $F$, described via the expanding cone in a maximal $\operatorname{Ad}$-diagonalizable subgroup of $F$ associated with horospherical $U$, allow us to prove similar thickness results via equidistribution. We will describe two types of restrictions we put on $F$, and to what extent these restrictions can be relaxed. As an application, we prove that the set of badly approximable matrices in the set-up of Diophantine approximation with quasimultiplicative weight functions has full Hausdorff dimension.
