Invariants of Bi-lipschitz Equivalence of Real Analytic Functions

We construct an invariant of the bi-Lipschitz equivalence of analytic function germs (Rn,0) → (R,0) that varies continuously in many analytic 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 coefficients of the asymptotic expansions of f along the sets where the size of |x||grad f(x)| is comparable to the size of |f (x)|. Consider a one parameter family of germs ft(x, y) : (R , 0) → (R, 0), t ∈ R, given by ft(x, y) = f(x, y, t) = x 3 − 3txy + y . (0.1) We shall show that if t 6= t, t, t > 0, then ft and ft′ are not bi-Lipschitz equivalent that is there is no germ of bi-Lipschitz homeomorphism h : (R, 0) → (R, 0) such that ft ◦h = ft′ . This shows in particular that the bi-Lipschitz classification of real analytic function germs admits continuous moduli. The existence of such moduli for complex analytic function germs was shown by the authors in [2]. We recall that, on the other hand, the bi-Lipschitz equivalence of complex or real analytic set germs does not admit moduli by [6] and [7], [8]. In order to distinguish bi-Lipschitz types of complex analytic function germs of two complex variables f : (C, 0) → (C, 0) we construct in [2] a numerical invariant that is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f . This construction is recalled in section 4 below. The real case, though similar, is more delicate. Firstly we cannot simply use the invariants of the complexification. For instance the family (0.1) for t < 0 is bi-Lipschitz trivial but the family of complexification admit continuous moduli. This is due to the fact that polar curves ∂f/∂x = 3(x − ty) = 0, t < 0, that are responsible for the complex invariant, are invisible in the reals. Given an analytic function germ f : (R, 0) → (R, 0). We associate to f a family of germs of sets (V (f), 0) ⊂ (R, 0) defined by the condition that |x||grad f(x)| is comparable to the size of |f(x)| on V (f), see sections 1 and 2 below. The sets V (f) are preserved by bi-Lipschitz equivalence and give rise to a numerical invariant, see section 3 and Theorem 5.1 below. This invariant is, in general, difficult to compute since it is not enough to use the branches of polar curve as in the complex case. Nevertheless, if n = 2, then one may use the complexification to simplify the computation, see Proposition 5 below. We compute some examples in section 6 and show, in particular, that in the family (0.1), t > 0, our invariant changes continuously. Notation and convention. We often write r instead of |x| which is the Euclidean norm of x. We use the standard notation φ = o(ψ) or φ = O(ψ) to compare the asymptotic behavior of φ and ψ, usually when we approach 0. We write φ ∼ ψ if 1991 Mathematics Subject Classification. 32S15, 32S05, 14H15. 1 2 JEAN-PIERRE HENRY AND ADAM PARUSIŃSKI φ = O(ψ) and ψ = O(φ), and φ ≃ ψ if φ ψ tends to 1. The gradient of a function f will be denoted by ∇f . 1. Characteristic exponents Given an analytic function germ f : (R, 0) → (R, 0). Following [5] we associate to f a finite set of positive rationals L(f) ⊂ Q, called the characteristic exponents of f , that are defined as follows. Let ∇f denote the gradient of f . The radial component of ∇f is defined on R \ {0} and equals