\(C ^{0}\) and bi-Lipschitz \({\mathcal{K}}\)-equivalence of mappings
type: Item
instance of: scholarly article
title: \(C ^{0}\) and bi-Lipschitz \({\mathcal{K}}\)-equivalence of mappings
zbMATH Open document ID: 1253.58017
author: Maria Aparecida Soares Ruas
author: Guillaume Valette
published in: Mathematische Zeitschrift
publication date: 2011-10-25T00:00:00Z
review text: The \(\mathcal{K}\)-equivalence (or contact equivalence) is an important notion in the study of smooth mappings. Geometrically, contact equivalence compares the singularities resulting from the intersection of the graphs of two germs having a singular point at the origin with the source axis. Whereas in the existing literature mostly \(C^\infty\)-\(\mathcal{K}\)-equivalence is investigated, the authors consider in the paper under review \(C^0\)-\(\mathcal{K}\)-equivalence and bi-Lipschitz \(\mathcal{K}\)-equivalence. We give the definitions. Definition. Two maps \(f,g:(\mathbb{R}^n;0)\to(\mathbb{R}^m;0)\) are \(C^0\)-\(\mathcal{K}\)-equivalent if there are germs of homeomorphisms \(h:(\mathbb{R}^n;0)\to(\mathbb{R}^n;0)\) and \(H:(\mathbb{R}^{n+m};0)\to(\mathbb{R}^{n+m};0)\) such that \[ \pi_n\circ H=h\circ\pi_n, \pi_m\circ H(x,0)=0\;\mathrm{ and }\;H\circ(\mathrm{id}_{\mathbb{R}^n},f)=(\mathrm{id}_{\mathbb{R}^n},g)\circ h, \] where \(\pi_n:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}^n\) is the orthogonal projection onto the first factor and \(\pi_m:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}^m\) is the orthogonal projection onto the second factor. The germs \(f\) and \(g\) are bi-Lipschitz \(\mathcal{K}\)-equivalent if \(h\) and \(H\) can be chosen bi-Lipschitz. Definition. Let \(F:U\times[0,1]\to\mathbb{R}^m\), \((x,t)\to F_t(x),\) be a deformation where \(U\) is an open neighbourhood of the origin in \(\mathbb{R}^n\). The deformation \(F\) is called \(C^0\)-\(\mathcal{K}\)-trivial if there exist mappings \(H:\mathbb{R}^{n+m}\times [0,1]\to\mathbb{R}^{n+m}\) and \(h:\mathbb{R}^n\times[0,1]\to\mathbb{R}^n\) such that for any \(t\) the pair \((h_t,H_t)\) is a \(C^0\)-\(\mathcal{K}\)-equivalence between \(F_0\) and \(F_t\). It is called bi-Lipschitz (resp. semialgebraically) \(\mathcal{K}\)-trivial if the homeomorphisms can be chosen bi-Lipschitz (resp. semialgebraic). The authors show results for the general case and for the semialgebraic case. In the general situati…
reviewed by: Tobias Kaiser
Mathematics Subject Classification ID: 58K40
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 32B10
zbMATH DE Number: 5963458
zbMATH Keywords: \(C^0\)-\(\mathcal{K}\)-equivalence
zbMATH Keywords: bi-Lipschitz \(\mathcal{K}\)-equivalence
zbMATH Keywords: \(\mathcal{K}\)-triviality
MaRDI profile type: Publication
full work available at URL: s00209-010-0728-z
OpenAlex ID: W2086760093
cites work: Topological \(\mathcal K\)-equivalence of analytic function-germs
cites work: $\mathcal{K}$-bi-Lipschitz equivalence of real function-germs
cites work: On the link of a stratum in a real algebraic set
cites work: Q4326414
cites work: Topological stability of smooth mappings
cites work: Q4805280
cites work: Lipschitz triangulations
DOI: 10.1007/S00209-010-0728-Z
description: scientific article; zbMATH DE number 5963458