Constructing bounded orbits of special types on homogeneous spaces

TL;DR

This paper investigates the structure of bounded orbits in homogeneous spaces, proving indecomposability of the set of points with precompact orbits and constructing points with large Hausdorff dimension orbit closures.

Contribution

It establishes the indecomposability of the set of points with precompact orbits and constructs points with orbit closures of arbitrarily large Hausdorff dimension in homogeneous spaces.

Findings

The set of points with precompact orbits is indecomposable.

For neutral subgroup dimension 1, many points have orbit closures with Hausdorff dimension close to the space.

The set of points with precompact orbits is dense and of full Hausdorff dimension.

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 $\dim X - \varepsilon$.