AI 中文总结
研究某些齐性空间乘积上锥作用的发散轨道，引入本质奇点概念，通过特定设置得出当\(\epsilon\in (0,\frac{1}{64})\)时本质发散轨道点集豪斯多夫维数的范围，扩展了前人在高维锥作用方面的结果。
AI 中文摘要
本文研究了某些齐性空间乘积上锥作用的发散轨道。引入了此类作用的本质奇点概念，并估计了相应奇异集的豪斯多夫维数。特别地，设\(G/\Gamma=\mathrm{SL}(2,\mathbb{R})^s/\mathrm{SL}(2,\mathbb{Z})^s\)，\(C\)是正Weyl腔中孔径为\(\epsilon>0\)的锥。当\(\epsilon\in (0,\frac{1}{64})\)时，本质发散轨道点集的豪斯多夫维数满足\(3s-\frac{1}{2}-4(s-1)\epsilon \leq \dim D^e(C, G/\Gamma)\leq 3s-\frac{1}{2}-\frac{1}{3}\epsilon\)，将前人结果扩展到了高维锥作用。
英文摘要
In this paper, we investigate divergent orbits for cone actions on products of certain homogeneous spaces. We introduce a notion of essential singularity for such actions, and estimate the Hausdorff dimension of the corresponding singular set. In particular, let $G/Γ=\mathrm{SL}(2,\mathbb{R})^s/\mathrm{SL}(2,\mathbb{Z})^s$, and let $C$ be a cone in the positive Weyl chamber with angular aperture $ε>0$. Then the Hausdorff dimension of the set of points with essential divergent orbits under $C$ satisfies that when $ε\in (0,\frac{1}{64})$, $$ 3s-\frac{1}{2}-4(s-1)ε\leq \dim D^e(C, G/Γ)\leq 3s-\frac{1}{2}-\frac{1}{3}ε. $$ This extends the previous result of An--Guan--Marnat--Shi \cite{AGMS} to higher-dimensional cone actions.
