Sequence of induced Hausdorff metrics on Lie groups

Abstract

Let $\phi :G\times (M,d)\to (M,d)$ 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 $gp:=\phi (g,p)$. 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 $d_X(g{H}_X,h{H}_X)=d_H(gX,hX)$, where $d_H$ is the Hausdorff distance in $M$. Suppose that $\phi$ is transitive and that there exist $p\in M$ such that $H_X=H_p$. Then $g{H}_X\mapsto gp$ 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 $\phi :G\times (M\equiv G/H_X,d^1)\to (M\equiv G/H_X,d^1)$, in order to get $d^2,d^3$ and so on. We study the particular case where $M=G$, the left action $\phi :G\times (G,d)\to (G,d)$ 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.

18 pages

References in corpus (1)

• Induced Hausdorff metrics on quotient spaces

Cited by in corpus (1)

• Mollifier smoothing of $C^0$-Finsler structures