Abstract
LetGbe a connected semisimple real Lie group,Γan irreducible lattice inGandX = G/Γ . LetF = {g_(t): t≥ 0}be a non-quasiunipotent one-parameter subsemigroup ofG . Then it is known that the set of points inXwith boundedF -trajectories has full Hausdorff dimension. In addition, ifUis the expanding horospherical subgroup relative tog₁ , then for anyx ∈ Xthe set of pointsu ∈ Usuch that theF -trajectory ofuxis bounded has full Hausdorff dimension. In this paper we takeUto be a horospherical subgroup ofGand apply Shi's equidistribution theorem for elements of the expanding cone with respect toUto describe a class of subsetsFinG , not presupposing the group structure, for which the above full Hausdorff dimension statements also hold. As an application, we prove that the set of badly approximable matrices in the set-up of Diophantine approximations with quasimultiplicative weight functions has full Hausdorff dimension.