Abstract
Let$X = G/Γ$be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup$F$of$G$that is$\operatorname{Ad}$ -diagonalizable over$\mathbb{C}$and whose action on$(X,m_X)$is mixing. In this dynamical system we study the set of points$x \in X$with a precompact orbit, written as$E(F,\infty)$ , which is known to be a dense subset of$X$of full Hausdorff dimension. We prove that$E(F,\infty)$is indecomposable in the following sense: given any$y \in E(F,\infty)$ , the set of$x \in E(F,\infty)$for which$y \in \overline{F_+x}$ , where$F_+$denotes the positive ray in$F$ , is uncountable and dense in$E(F,\infty)$ . When the dimension of the neutral subgroup of$G$with respect to$F$is$1$we demonstrate, for any$\varepsilon>0$ , the existence of many points$x \in X$whose orbit closure$\overline{F_+x} \subset X$is compact and has Hausdorff dimension at least$\dim X - \varepsilon$ .