Mather proved that the smooth stability of smooth maps between manifolds is a generic condition if and only if the pair of dimensions of the manifolds are ‘nice dimensions’ while topological stability is a generic condition in any pair of dimensions. And, by a result of du Plessis and Wall $C^1$-stability is also a generic condition precisely in the nice dimensions. We address the question of bi‑Lipschitz stability in this article. We prove that the Thom–Mather stratification is bi‑Lipschitz contact invariant in the boundary of the nice dimensions. This is done in two steps: first we explicitly write the contact unimodular strata in every pair of dimensions lying in the boundary of the nice dimensions and second we construct Lipschitz vector fields whose flows provide the bi‑Lipschitz contact trivialization in each of the cases.

中文翻译：

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精密尺寸边界内的接触轨道的双里普希兹几何

马瑟（Mather）证明，当且仅当歧管的成对尺寸为“ nice Dimensions”，而拓扑稳定性为任意一对尺寸的通配条件时，流形之间的平滑映射的平滑稳定性才是通用条件。而且，由于du Plessis和Wall的作用，$ C ^ 1 $-稳定性也恰好是良好尺寸的通用条件。在本文中，我们将解决bi-Lipschitz稳定性问题。我们证明了Thom-Mather分层在美观维度的边界上是bi-Lipschitz接触不变的。这分两个步骤完成：首先，在漂亮尺寸的边界内，在每对尺寸中显式地编写接触单模地层；其次，我们构造Lipschitz矢量场，其流在每种情况下均提供bi-Lipschitz接触琐事。