李群上的诱导Hausdorff度量序列
Sequence of induced Hausdorff metrics on Lie groups
摘要：
设$φ: G \times (M,d) \rightarrow (M,d)$是李群在可微流形上的左作用，该流形具有与$M$拓扑相容的度量$d$（距离函数）。表示为$gp:=φ(g,p)$。令$X$是$M$的紧致子集。则$X$的各向同性子群是$G$的闭子群，定义为$H_X:=\{g\in G; gX=X\}$。导出的Hausdorff度量是左陪集流形$G/H_X$上的度量，定义为$d_X(gH_X,hH_X)=d_H(gX,hX)$，其中$d_H$是$M$中的Hausdorff距离。假设$φ$是传递的，并且存在$p\in M$使得$H_X=H_p$。则$gH_X \mapsto gp$是一个标识$G/H_X$和$M$的同构。在这项工作中，我们定义了一个离散动力系统的$M$度量。令$d^1=\hat d_X$，其中$\hat d_X$代表与$d_X$相关联的固有度量。研究了$φ: G \times (M\equiv G/H_X,d^1)\rightarrow (M\equiv G/H_X,d^1)$，左作用$φ: G\times (G,d) \rightarrow (G,d)$是$G$的乘积，$d$被$G$上的一个右不变内度量上有界，$X\ni e$是$G$的一个有限子集的特殊情形.我们证明了序列$d^i$逐点收敛到度量$d^\infty$。另外，如果$d$是完备的，并且由$X$生成的半群在$G$中稠密，则$d^\infty$是右不变$C^0$-Carnot-Carathéodory-Finsler度量的距离函数。详细研究了$d^\infty$是$C^0$-Finsler的情形.
英文摘要：
Let $φ: G \times (M,d) \rightarrow (M,d)$ be a left action of a Lie group on a differentiable manifold endowed with a metric $d$ (distance function) compatible with the topology of $M$. Denote $gp:=φ(g,p)$. Let $X$ be a compact subset of $M$. Then the isotropy subgroup of $X$ is a closed subgroup of $G$ defined as $H_X:=\{g\in G; gX=X\}$. The induced Hausdorff metric is a metric on the left coset manifold $G/H_X$ defined as $d_X(gH_X,hH_X)=d_H(gX,hX)$, where $d_H$ is the Hausdorff distance in $M$. Suppose that $φ$ is transitive and that there exist $p\in M$ such that $H_X=H_p$. Then $gH_X \mapsto gp$ is a diffeomorphism that identifies $G/H_X$ and $M$. In this work we define a discrete dynamical system of metrics on $M$. Let $d^1=\hat d_X$, where $\hat d_X$ stands for the intrinsic metric associated to $d_X$. We can iterate $φ: G \times (M\equiv G/H_X,d^1)\rightarrow (M\equiv G/H_X,d^1)$, in order to get $d^2, d^3$ and so on. We study the particular case where $M=G$, the left action $φ: G\times (G,d) \rightarrow (G,d)$ is the product of $G$, $d$ is bounded above by a right invariant intrinsic metric on $G$ and $X\ni e$ is a finite subset of $G$. We prove that the sequence $d^i$ converges pointwise to a metric $d^\infty$. In addition, if $d$ is complete and the semigroup generated by $X$ is dense in $G$, then $d^\infty$ is the distance function of a right invariant $C^0$-Carnot-Carathéodory-Finsler metric. The case where $d^\infty$ is $C^0$-Finsler is studied in detail.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
