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- W2593759543 abstract "Let $varphi: G times (M,d) rightarrow (M,d)$ 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 $gp:=varphi(g,p)$. Let $X$ be a compact subset of $M$. Then the isotropy subgroup of $X$ is a closed subgroup of $G$ defined as $H_X:={gin G; gX=X}$. 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 $varphi$ is transitive and that there exist $pin M$ 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 $varphi: G times (Mequiv G/H_X,d^1)rightarrow (Mequiv G/H_X,d^1)$, in order to get $d^2, d^3$ and so on. We study the particular case where $M=G$, the left action $varphi: Gtimes (G,d) rightarrow (G,d)$ is the product of $G$, $d$ is bounded above by a right invariant intrinsic metric on $G$ and $Xni e$ 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-Carath'eodory-Finsler metric. The case where $d^infty$ is $C^0$-Finsler is studied in detail." @default.
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- W2593759543 date "2019-05-25" @default.
- W2593759543 modified "2023-10-03" @default.
- W2593759543 title "Sequence of Induced Hausdorff Metrics on Lie Groups" @default.
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- W2593759543 doi "https://doi.org/10.1007/s00574-019-00151-2" @default.
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