Bilipschitz determinacy of quasihomogeneous germs

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1 CADERNOS DE MATEMÁTICA 03, April (2002) ARTIGO NÚMERO SMA#139 Bilipschitz determinacy of quasihomogeneous germs Alexandre Cesar Gurgel Fernandes * Centro de Ciências,Universidade Federal do Ceará Av. Humberto Monte, s/n Campus do Pici - Bloco 914 Fortaleza,Ceará,Brazil e Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, São Carlos SP, Brazil alex@icmc.sc.usp.br and Maria Aparecida Soares Ruas Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, São Carlos SP, Brazil maasruas@icmc.sc.usp.br We obtain estimates for the degree of bilipschitz determinacy of quasihomogeneous function-germs. April, 2002 ICMC-USP 1. INTRODUCTION A basic problem in Singularity Theory is the local classification of mappings module diffeomorphisms. In 1965, H. Whitney justified the rigidity of the classification problem by C 1 -diffeomorphism giving the following example: F t (x, y) = xy(x y)(x ty); 0 < t < 1 (1) which presents the following phenomenon: for any t s in I = (0, 1) it is not possible to construct a C 1 -diffeomorphism φ : (R 2, 0) (R 2, 0) such that F t = F s φ. This motivated the classification of mappings by isomorphisms weaker than diffeomorphisms. There is an extensive literature related to C r -equivalence (1 r < ) of map-germs, among them [5], [4] and [1] which are more closely related to this work. However, only few * Research partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico, (CNPq), Brazil, grant # /00-8. Research partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico, (CNPq), Brazil, grant # /88-0, and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil, grant # 97/ Sob a supervisão CPq/ICMC

2 116 A. FERNANDES AND M. RUAS recent works deal with the problem of bilipschitz classification of map-germs. This work is inspired in a recent paper by J.-P. Henry and A. Parusinski [2], where they show that the bilipschitz equivalence of analytic function-germs admits continuous moduli. We obtain estimates for the degree of bilipschitz determinacy of quasihomogeneous function-germs. Examples are given to show that the estimates are sharp. 2. BILIPSCHITZ EQUIVALENCE Let λ R be a positive number. A mapping φ : U R n R p is called λ-lipschitz, or simply Lipschitz if it satisfies: φ(x) φ(y) λ x y x, y U. When n = p and φ has a Lipschitz inverse, we say that φ is bilipschitz. Two germs f, g : (R n, 0) (R p, 0) are called bilipschitz equivalent if there exists a bilipschitz map-germ φ : (R n, 0) (R n, 0) such that f = g φ. Example Let f, g : (R, 0) (R, 0) be given by f(x) = x, g(x) = x 3. It is easy to show that f and g are not bilipschitz equivalent. On the other hand, there is a homeomorphism φ : (R, 0) (R, 0) such that f = φ g. Let f : (R n, 0) (R, 0) be the germ of an analytic function, f(x) = f m (x) + f m+1 (x) +, with f i a homogeneous form of degree i, and f m 0. We denote by m f := m, the multiplicity of f. We say that f has non-degenerate tangent cone if 0 R n is the only point in R n in which f m x 1 = = f m x 1 = 0. Proposition Let f : (R n, 0) (R, 0) be the germ of an analytic function. Then m f = ord r [sup f B(0,r) ], where B(0, r) denote the ball centered at the origin with radius r. Proof. Let α = ord r [sup f B(0,r) ]. Write f(x) = f m (x) + f m+1 (x) + Sob a supervisão da CPq/ICMC

3 BILIPSCHITZ DETERMINACY OF QUASIHOMOGENEOUS GERMS 117 with f i a homogeneous form of degree i, and f m 0. Let x = (x 1,..., x n ) be such that f m (x) 0. Then, given r > 0 we have f(rx) = r m f m (x) + rf m+1 (x) + Kr m for some constant K > 0, hence m α. On the other hand, from the Curve Selection Lemma, there exists an analytic arc γ : [0, ɛ) R n, γ(0) = 0, such that α = ord r f(γ(r) and γ(r) r for each r > 0. Since γ(0) = 0, we can write γ(r) = r γ(r) with lim r 0 γ(r) <. Therefore, f(γ(r)) = r m f m ( γ(r)) + rf m+1 ( γ(r)) + Lr m for some constant L > 0. Hence, m α. Corollary Let f, g : (R n, 0) (R, 0) be germs of analytic functions. If f and g are bilipschitz equivalent, then m f = m g. The corollary above in the complex case was proved by J.-J. Risler and D. Trotman in [3]. It is obvious that the converse statement is false, but we can prove the following result Proposition Let F t : (R n, 0) (R, 0) be a smooth family of smooth functiongerms. If m Ft is constant and F t has non-degenerate tangent cone for each t, then for each t s, F t and F s are bilipschitz equivalent. The result above will follow as consequence of Theorem Corollary The family (1) satisfies: F t and F s are bilipschitz equivalent t, s (0, 1). It is valuable to observe that the Proposition does not guarantee the non-rigidity of the bilipschitz classification problem for analytic functions. In fact, J.-P. Henry and A. Parusinski ([2]) presented the family F t : (C 2, 0) (C 2, 0) given by F t (x, y) = x 3 3t 2 xy 2 + y 6 which satisfies: for any t s (0, 1 2 ) there is no bilipschitz map-germ φ : (C2, 0) (C 2, 0) such that F t = F s φ. The proof is based on the analysis of the expansion of the germs of the family along each arc of their polar curves. The argument in [2] also holds in the real case, that is, the following holds: Sob a supervisão CPq/ICMC

4 118 A. FERNANDES AND M. RUAS Proposition The family F t : (R 2, 0) (R 2, 0) given by F t (x, y) = x 3 3t 2 xy 2 + y 6 satisfies: for any t s (0, 1 2 ) there is no bilipschitz map φ : (R2, 0) (R 2, 0) such that F t = F s φ. Note that F t is a deformation of the quasihomogeneous germ f = x 3 + y 6 which has an isolated singularity at origin. Therefore, it is natural to ask for which θ(x, y) the family f + tθ is bilipschitz trivial. 3. BILIPSCHITZ DETERMINACY OF QUASIHOMOGENEOUS GERMS Let f t : (R n, 0) (R, 0) t I (an interval in R), be a smooth family of smooth function-germs. That is, there is a neighborhood U of 0 in R n and a smooth function F : U I R such that F (0, t) = 0 and f t (x) = F (x, t) t I, x U. We call f t strongly bilipschitz trivial when there is a continuous family of λ-lipschitz map-germs (vector field) v t : (R n, 0) (R n, 0) such that t R and x near 0 in R n. f t t (x) = (df t) x (v t (x)) Theorem If f t is bilipschitz trivial, then for each t s I there is a bilipschitz map-germ φ : (R n, 0) (R n, 0) such that f t = f s φ. The theorem above is known as a result of Thom-Levine type and its proof is immediate, since the flow of a Lipschitz vector field is bilipschitz. Let E n be the space of smooth function-germs (R n, 0) R. Given f E n, we denote Nf(x) = [ f (x)] 2. We say that Nf(x) satisfies a Lojasiewicz condition if there exist constants c > 0 and α > 0 such that Nf(x) c x α. Fix the weights (r 1,..., r n ). We recall that a function f is called quasihomogeous with respect to the given weights if there is a number d such that f satisfies the following equation: f(λ x) = λ d (x 1,..., x n ) λ R and x = (x 1,..., x n ) R n, where λ x = (λ r 1 x 1,..., λ r n x n ). With respect to the given weights, for each monomial x α = x α 1 1 xαn n, we define fil(x α ) = n i=1 α ir i. We define a filtration in the ring E n via the function fil(f) = min{fil(x α ) : ( f x )(0) 0}, α for each f E n. We can extend this definition to E n+1, the ring of 1-parameter families of smooth function-germs in E n, by defining fil(x α t β ) = fil(x α ). Let (r 1,..., r n ; 2k) be fixed. The standard control function of type (r 1,..., r n ; 2k) is ρ(x) = x 2α x 2α n, where the α i are chosen such that the function ρ is quasihomogeneous of type (r 1,..., r n ; 2k). Sob a supervisão da CPq/ICMC

5 BILIPSCHITZ DETERMINACY OF QUASIHOMOGENEOUS GERMS 119 Lemma Let h(x) be a quasihomogeneous polynomial function of type (r 1,..., r n ; 2k), with r 1 r n, ρ the standard control function of same type that h and h t (x) a deformation of h such that: fil(h t ) 2k + r n, t [0, 1]. (2) Then the function h t(x) ρ(x) is c-lipschitz, with c independent of t. Proof. Without loss of generality, we can suppose that h t (x) is quasihomogeneous of type (r 1,..., r n ; d) where d 2k + r n. We consider G t (x, y) : R n R n R n given by G t (x, y) = ρ(y)h t (x) ρ(x)h t (y), m t (x, y) = x y ρ(x)ρ(y) and M = {(x, y, t) : G t (x, y) = 1}. Since M is closed, the number c = inf{m t (x, y) : (x, y, t) M)} is positive. Now, let x, y R n be sufficiently near the origin x 0, y 0 and x y. Let λ > 0 be such that G t (λ x, λ y) = 1, that is, On the other hand, we use that λ > 1 to obtain: G t (x, y) = 1 λ 2k+d (3) m t (λ x, λ y) = λ 4k λ x λ y ρ(x)ρ(y) λ 4k+r n x y ρ(x)ρ(y) = λ 4k+r n m t (x, y) 1 m t (x, y) c. (4) 4k+rn λ Now, we use that λ > 1, d 2k + r n, (3), (3) and we obtain the following inequality m t (x, y) cg t (x, y), that is, h t(x) ρ(x) h t(y) ρ(y) c 1 x y. Theorem Let f : R n, 0 R n, 0 be the germ of a quasihomogeneous polynomial function of type (r 1,..., r n ; d), r 1 r n with isolated singularity. Let f t (x) = f(x) + tθ(x, t), t [0, 1], be a smooth deformation of f. If fil(θ) d + r n r 1, then f t admits a strong bilipschitz trivialization along I = [0, 1]. Proof. We can see that for each i there exists a s i such that f of the type (r 1,..., r n ; s i ), s i = d r i. is quasihomogeneous Sob a supervisão CPq/ICMC

6 120 A. FERNANDES AND M. RUAS Let N f be defined by N f = [ ] 2αi f, where α i = k s i and k = l.c.m.(s i ). Therefore N f is a quasihomogeneous control function of the type (r 1,..., r n ; 2k). The lemma bellow is proved in [4]. Lemma There exist constants 0 < c 2 < c 1 such that c 2 ρ(x) N f t (x) c 1 ρ(x). We have the following equation; where W is given by f t t N f t = df t (W ), W = W i where W i = f [ ] 2αi 1 t f. t ( ) Since fil ft t d + r n r 1 and ( [ ] ) 2αi 1 ( ) ft ft fil = (2α i 1)fil = (2α i 1)(d r i ) = 2K d + r i 2k d + r 1 we have that min fil(w i ) fil(θ) + 2k d + r 1 2k + r n. Let v : R n R, 0 R n W R, 0 be the vector field given by NR f. From Lemma 3.3.2, it t follows that v is a Lipschitz vector field. Finally the equation ( ft t )(x) = (df t) x (v(x, t)) gives the strong bilipschitz triviality of the family f t (x) along a small open interval around t = 0. Since the same argument is true for each t I, the proof is complete. The following result shows that the estimate given in Theorem is sharp. Proposition Let f t : (R 2, 0) (R, 0); t I = ( δ, δ) R be given by f t (x, y) = 1 3 x3 t 2 xy 3n 2 + y 3n. Sob a supervisão da CPq/ICMC

7 BILIPSCHITZ DETERMINACY OF QUASIHOMOGENEOUS GERMS 121 Then f t is not strongly bilipschitz trivial. Remark Let f(x, y) = 1 3 x3 + y 3n. Note that f is quasihomogeneous of type (n, 1; 3n). From Theorem it follows that f + tθ is strongly bilipschitz trivial for each θ(x, t) such that fil(θ) 4n 1. Proof (of the Proposition 3.3.5). Let m = 3n 2. Here we repeat the argument-proof from Theorem 1.1 in [2]. Suppose that v(x, y, t) = v 1 (x, y, t) x + v 2(x, y, t) y is a vector field such that: ( f t t )(x, y) = (df t) x (v(x, y, t)) The polar curve of f t {(x, y) R 2 f t : x (x, y) = 0} is equal to the set {(x, y) R2 : x 2 = t 2 y m }. Thus, v 1 (ty m/2, y, t) and v 2 ( ty m/2, y, t) satisfy: v 1 (ty m/2, y, t) f t y (tym/2, y, t) = f t t (tym/2, y, t) (5) v 2 ( ty m/2, y, t) f t y ( tym/2, y, t) = f t t ( tym/2, y, t). (6) From equations (5) and (6) we have: v 1 (ty m/2, y, t) = 2t 2 y m/2 1 mt 3 y m/ n Thus, On the other hand, v 2 ( ty m/2, y, t) = 2t2 y m/2 1 mt 3 y m/ n v 1 (ty m/2, y, t) v 2 ( ty m/2, y, t) y m/2 1 (7) (ty m/2, y, t) ( ty m/2, y, t) y m/2 (8) But, (7) and (8) show that v 2 is not Lipschitz. Hence f is not strongly bilipschitz trivial. The invariant for bilipschitz equivalence Inv(f t ) presented in [2] is independent of t, hence does not distinguish the element of the given family f t. Sob a supervisão CPq/ICMC

8 122 A. FERNANDES AND M. RUAS REFERENCES 1. S. Bromberg and S. L. de Medrano, C r -sufficiency of quasihomogeneous functions. Real and Complex Singularities, Pitman Research Notes in Mathematics Series 333, (1995), A. Parusinski and J.-P. Henry, Existence of moduli for bilipschitz equivalence of analytic functions. Prépublications de l Univesité d Angers no. 137, J.-J. Risler and D. Trotman, bilipschitz invariance of the multiplicity. Bull. London Math. Soc. 29 (1997), no. 2, M.A.S. Ruas and M.J. Saia, C l -determinacy of weighted homogeneous germs. Hokkaido Mathematical Journal, vol. 26, no 1 (1997), C. T. C. Wall, Finite determinacy of smooth map-germs. Bull. London Math. Society 13 (1981), Sob a supervisão da CPq/ICMC