Constructing bounded orbits of special types on homogeneous spaces

Abstract

Let $X = G/\Gamma$ 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 $\dim X - \varepsilon$.