\(C ^{0}\) and bi-Lipschitz \({\mathcal{K}}\)-equivalence of mappings
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 situation they prove the following. Theorem. Let \(F:U\times[0,1]\to \mathbb{R}^m\) be a \(C^0\) deformation. If \(F_t^{-1}(0)\) is locally topologically trivial at the origin then \(F\) is \(C^0\)-\(\mathcal{K}\)-trivial. Moreover, they give a new proof of a degree criterion for \(C^0\)-\(\mathcal{K}\)-equivalence of \(C^\infty\)-germs by \textit{T. Nishimura} [Paris: Éditeurs des Sciences et des Arts. Trav. Cours. 54, 83--93 (1997; Zbl 0896.58008)], using the second Thom-Mather's isotopy theorem instead of the Poincaré conjecture. In the semialgebraic situation they show a semialgebraic version of the above theorem, avoiding integration of vector fields in the proof. Theorem. Let \(F:U\times[0,1]\to \mathbb{R}^m\) be a semialgebraic deformation. If \(F_t^{-1}(0)\) is semialgebraically topologically trivial at the origin then \(F\) is semialgebraically \(C^0\)-\(\mathcal{K}\)-trivial. For polynomial deformations a criterion for semialgebraic bi-Lipschitz \(\mathcal{K}\)-triviality is given involving the notion of the real spectrum. Given a polynomial family of maps the members are piecewise bi-Lipschitz \(\mathcal{K}\)-equivalent.
bi-Lipschitz \(\mathcal{K}\)-equivalence
scientific article; zbMATH DE number 5963458