C 0 and bi-Lipschitz {\mathcal{K}} -equivalence of mappings

In this paper we investigate the classification of mappings up to ${\mathcal{K}}$ -equivalence. We give several results of this type. We study semialgebraic deformations up to semialgebraic C ⁰ ${\mathcal{K}}$ -equivalence and bi-Lipschitz ${\mathcal{K}}$ -equivalence. We give an algebraic criterion for bi-Lipschitz ${\mathcal{K}}$ -triviality in terms of semi-integral closure (Theorem 3.5). We also give a new proof of a result of Nishimura: we show that two germs of smooth mappings ${f, g: \mathbb{R}^n \to \mathbb{R}^n}$ , finitely determined with respect to ${\mathcal{K}}$ -equivalence are C ⁰- ${\mathcal{K}}$ -equivalent if and only if they have the same degree in absolute value.