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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.16095 |
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| _version_ | 1866918316792086528 |
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| author | Einsiedler, Manfred Kleinbock, Dmitry Rao, Anurag |
| author_facet | Einsiedler, Manfred Kleinbock, Dmitry Rao, Anurag |
| contents | 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 $\dim X - \varepsilon$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16095 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing bounded orbits of special types on homogeneous spaces Einsiedler, Manfred Kleinbock, Dmitry Rao, Anurag Dynamical Systems 37A17, 37A25, 37D40, 11J70 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 $\dim X - \varepsilon$. |
| title | Constructing bounded orbits of special types on homogeneous spaces |
| topic | Dynamical Systems 37A17, 37A25, 37D40, 11J70 |
| url | https://arxiv.org/abs/2511.16095 |