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Existence of moduli for bi-Lipschitz equivalence of analytic functions. (English) Zbl 1026.32055

Let \(f_t(x,y) = x^3 - 3t^2 x y^4 + y^6\) define a one parameter family of germs \(f_t : (\mathbb{C}^2,0) \to (\mathbb{C},0)\), \(t \in \mathbb{C}\). It is proved that if \(t,t^\prime\) are sufficiently generic, then \(f_t\) and \(f_{t^\prime}\) are not bi-Lipschitz equivalent function germs, that is, there is no germ of bi-Lipschitz homeomorphism \(H : (\mathbb{C}^2,0) \to (\mathbb{C}^2,0)\) such that \(f_t \circ H = f_{t^\prime}\).

MSC:

32S05 Local complex singularities
14H15 Families, moduli of curves (analytic)
32S15 Equisingularity (topological and analytic)
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