Compositio Mathematica

Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions

Jean-Pierre Henry a1 and Adam Parusinski a2
a1 Centre de Mathématiques, (Unité associé au CNRS No169), Ecole Polytechnique, F-91128 Palaiseau Cedex, France. e-mail:
a2 Département de Mathématiques, Université d'Angers, 2, bd Lavoisier, 49045 Angers Cedex 1, France. e-mail:

Article author query
henry jp   [Google Scholar] 
parusinski a   [Google Scholar] 


We show that the bi-Lipschitz equivalence of analytic function germs (${\open C}^{2}$, 0)[rightward arrow](${\open C}$, 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 ft: (${\open C}^{2}$, 0)[rightward arrow](${\open C}$, 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.

Key Words: analytic function germs; bi-Lipschitz equivalence; moduli; polar curve; Newton polygon.