Constructing bounded orbits of special types on homogeneous spaces

Abstract

Let $X=G/\hat{I}“$ be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup $F$ of $G$ that is $\mathrm{Ad}$-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 $\epsilon >0$, the existence of many points $x\in X$ whose orbit closure $\overline{F_{+}x}\subset X$ is compact and has Hausdorff dimension at least $\dim X-\epsilon$.

18 pages; several misprints were corrected and the presentation was improved