Abstract
We show that the bi-Lipschitz equivalence of analytic function germs (ℂ2, 0)→(ℂ, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families f t : (ℂ2, 0)→(ℂ, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.
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Henry, JP., Parusiński, A. Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions. Compositio Mathematica 136, 217–235 (2003). https://doi.org/10.1023/A:1022726806349
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DOI: https://doi.org/10.1023/A:1022726806349