Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices
Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices
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DOI：
10.3934/dcds.2020227
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发表时间：
2016-06
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影响因子：
--
通讯作者：
N. Shah;Lei Yang
中科院分区：
文献类型：
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作者：
N. Shah;Lei Yang
本文研究了一条解析曲线 \begin{document}$ \varphi: I = [a, b]\rightarrow \mathrm{M}(m\times n, \mathbb{R}) $\end{document} 在 \begin{document}$ m $\end{document} 乘以 \begin{document}$ n $\end{document} 实矩阵的空间中，并证明如果 \begin{document}$ \varphi $\end{document} 满足一定的几何条件，那么对于曲线上的几乎每个点，狄利克雷定理给出的丢番图近似都无法改进。为此，我们将曲线嵌入到齐次空间 \begin{document}$ G/\Gamma $\end{document} 中，并证明在某些扩展对角子群 \begin{document}$ A = \{a(t): t \in \mathbb{R}\} $\end{document} 的作用下，曲线的平移往往在 \begin{document}$ G/\Gamma $\end{document} 中均匀分布，如 \begin{document}$ t \rightarrow +\infty $\end{document} 。证明依赖于线性化技术和表示理论。
In this paper, we study an analytic curve \begin{document}$ \varphi: I = [a, b]\rightarrow \mathrm{M}(m\times n, \mathbb{R}) $\end{document} in the space of \begin{document}$ m $\end{document} by \begin{document}$ n $\end{document} real matrices, and show that if \begin{document}$ \varphi $\end{document} satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into a homogeneous space \begin{document}$ G/\Gamma $\end{document} , and prove that under the action of some expanding diagonal subgroup \begin{document}$ A = \{a(t): t \in \mathbb{R}\} $\end{document} , the translates of the curve tend to be equidistributed in \begin{document}$ G/\Gamma $\end{document} , as \begin{document}$ t \rightarrow +\infty $\end{document} . The proof relies on the linearization technique and representation theory.