Parusiński : Invariants of bi-Lipschitz equivalence of real analytic functions

Abstract. We construct an invariant of the bi-Lipschitz equivalence of an-alytic function germs ( n;0) ! (;0) that varies continuously in many ana-lytic families. This shows that the bi-Lipschitz equivalence of analytic function germs admit continuous moduli. For a germ f the invariant is given in terms of the leading coecients of the asymptotic expansions of f along the sets where the size of jxjjgrad f(x)j is comparable to the size of jf (x)j. Consider a one parameter family of germs ft(x; y) : (R 2; 0) ! (R; 0), t 2 R, given by ft(x; y) = f(x; y; t) = x 3 3txy4 + y6:(0.1) We shall show that if t 6 = t0, t; t0> 0, then ft and ft0 are not bi-Lipschitz equivalent that is there is no germ of bi-Lipschitz homeomorphism h: (R2; 0) ! (R2; 0) such that ft h = ft0. This shows in particular that the bi-Lipschitz classication of rea