Constructing bounded orbits of special types on homogeneous spaces
Manfred Einsiedler, Dmitry Kleinbock, Anurag Rao
TL;DR
The paper analyzes bounded (precompact) trajectories for a mixing Ad-diagonalizable one-parameter flow on X = G/Γ with non-uniform lattice Γ. By reducing the problem to the expanding subgroup H and exploiting uniform equidistribution on H, it constructs Cantor-like, tree-structured sets of H-elements that yield large, fractal-like sets of points with bounded F_+-orbits and prescribed limit behavior. The main contributions include an indecomposability result for E(F,∞): for any y ∈ E(F,∞) the set of x ∈ E(F,∞) with y in the closure of F_+ x is uncountable and dense, and, under dim Z = 1, the existence of many x with compact orbit closures of Hausdorff dimension at least dim X − ε. These results extend our understanding of bounded trajectories in homogeneous dynamics beyond measure and generic-dimension statements, using a Cantor-type construction grounded in expanding-subgroup dynamics.
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 $\dim X - \varepsilon$.