Parabolic curves for diffeomorphisms in C2

Publ. Mat. 52 (2008), 189–194 PARABOLIC CURVES FOR DIFFEOMORPHISMS IN C2 F. E. Brochero Mart´ ınez, F. Cano, and L. L´ opez-Hernanz Abstract We give a simple proof of the existence of parabolic curves for diffeomorphisms in (C2 ,0) tangent to the identity with isolated fixed point. 1. Introduction Let Fbe a diffeomorphism of (Cn,0) tangent to the identity. A parabolic curve for Fis an injective holomorphic map ϕ: Ω →Cn, where Ω is a simply connected domain in Cwith 0 ∈∂Ω such that (1) ϕis continuous at the origin, and ϕ(0) = 0. (2) F(ϕ(Ω)) ⊂ϕ(Ω) and F◦k(p) converges to 0 when k→+∞, for p∈ϕ(Ω). We say that ϕis tangent to [v]∈Pn−1if [ϕ(ζ)] →[v] when ζ→0. Let us write F(z) = z+Pk(z)+Pk+1(z)+···, where Pjis a n-dimensional vector of homogeneous polynomials of degree j, and Pk6≡ 0. A characteristic direction for Fis a point [v]∈Pn−1such that Pk(v)=λv, for some λ∈C; it is nondegenerate if λ6= 0. The integer ord(F) := k≥2 is the tangency order of Fat 0. The following theorem is analogous to Briot and Bouquet’s theorem [3] for diffeomorphisms of (Cn,0). Theorem 1.1 (Hakim [6]).Let Fbe a germ of diffeomorphism of (Cn,0) tangent to the identity. For any nondegenerate characteristic direction [v]there exist ord(F)−1disjoint parabolic curves tangent to [v] at the origin. When n= 2, Abate proved that the nondegeneracy condition can be dismissed. 2000 Mathematics Subject Classification. 32H02, 32H50, 37F99. Key words. Dynamical systems, discrete dynamics, parabolic curves, exponential map, blow-ups. The first author was supported by CAPES, Brazil, Process: BEX3083/05-5. The third author was supported by FPU program, Spain, Process:AP2005-3784. 190 F. E. Brochero Mart´ ınez, F. Cano, L. L´ opez-Hernanz Theorem 1.2 (Abate [1], [2]).Let Fbe a germ of diffeomorphism of (C2,0) tangent to the identity such that 0is an isolated fixed point. Then there exist ord(F)−1disjoint parabolic curves for Fat the origin. This theorem is analogous to Camacho-Sad’s theorem [4] of existence of invariant curves for holomorphic vector fields. We show in this note that the analogy is deep enough to prove Theorem 1.2 in a simple way starting with Hakim’s theorem. 2. Exponential operator and blow-up transformation Let b X2(C2,0) be the module of formal vector fields X=a(x, y)∂ ∂x + b(x, y)∂ ∂y of order ≥2, i.e., min{ν(a), ν(b)}≥2. We denote by d Diff1 (C2,0) the group of formal diffeomorphisms tangent to the identity F(x, y) = (x+p(x, y), y +q(x, y)) where min{ν(p(x, y)), ν(q(x, y)} ≥ 2. Let us denote by X2(C2,0) and by Diff1(C2,0) the convergent elements of b X2(C2,0) and d Diff1(C2,0) respectively. Let X∈b X2(C2,0). The exponential operator of Xis the application exp tX :C[[x, y]] →C[[x, y, t]] defined by the formula exp tX(g) = ∞ X j=0 tj j!Xj(g) where X0(g) = gand Xj+1(g) = X(Xj(g)). Note that, since ν(Xj(g)) ≥ j+ν(g), we can substitute t= 1 to get the element exp X(g)∈C[[x, y]]. Moreover, exp tX gives a homomorphism of C-algebras, in particular, we have exp tX(fg) = exp tX(f) exp tX(g). We get also Proposition 2.1. The application Exp: b X2(C2,0) →d Diff1(C2,0) X7→ (exp X(x),exp X(y)) is a bijection. Proof: Let G(x, y) = x+ ∞ P n=2 pn(x, y), y + ∞ P n=2 qn(x, y) and X= ∞ P n=2an(x, y)∂ ∂x +bn(x, y)∂ ∂y . Parabolic Curves for Diffeomorphisms in C2191 The identity Exp(X) = Gis equivalent to pm+1 =am+1 +HTm+1   m X j=2 1 j!Xj m(x)  qm+1 =bm+1 +HTm+1   m X j=2 1 j!Xj m(y) , where Xm= m P n=2an(x, y)∂ ∂x +bn(x, y)∂ ∂y ,and HTm+1(h) is the homogeneous term of hof order m+ 1. These equations determine univocally Xif Gis given. Note that ord(G) = ν(X). In general, Xmay not be convergent for a convergent G. The formal vector field Xsuch that G= Exp(X) is called the infinitesimal generator of G. If k=ν(X), then ak=pk and bk=qk, thus the characteristic directions of Fcorrespond to the points of the tangent cone of X. Moreover, if X=fX′with X′∈ b X(C2,0) and f∈C[[x, y]] then Exp(X)(x, y) = (x+f(x, y)p(x, y), y + f(x, y)q(x, y)). The converse statement follows by a process similar to the proof of Proposition 2.1. In particular, 0 is an isolated singular point of Xif and only if 0 is an isolated fixed point of F. Now, let π: (M, D)→(C2,0) be the blow up of C2at the origin, where D=π−1(0) = P1, thus each characteristic direction determines a point of D. Proposition 2.2. Let F∈Diff1(C2,0). There exists a unique germ of diffeomorphism ˜ Fin (M, D)such that π◦˜ F=F◦πand ˜ F|D= id|D. Moreover, the germ ˜ Fphas order ≥ord(F)for any characteristic direction p∈Dand hence ˜ Fp∈Diff1(M, p). Proof: Let F(x, y) = (x+pk(x, y) + ··· , y +qk(x, y) + ···) where k= ord(F)≥2. We have two charts of M=U1∪U2such that π|U1:U1→ C2, is defined by π(x, v) = (x, xv) and π|U2:U2→C2, is defined by π(u, y) = (uy, y). We define ˜ Fin the first chart as ˜ F(x, v) =π−1◦F◦π(x, v)=x+pk(x, xv) + ··· ,vx +qk(x, xv) + ··· x+pk(x, xv) + ···  =(x+xk(pk(1, v)+x(···)), v+xk−1(qk(1, v)−vpk(1, v)+x(···))). 192 F. E. Brochero Mart´ ınez, F. Cano, L. L´ opez-Hernanz Observe that ˜ F(0, v) = (0, v), thus any point of the divisor is fixed. Moreover, if qk(1, v0)−v0pk(1, v0) = 0 we have dF(0, v0) = I, and thus for any characteristic direction p= (0, v0)∈D, ord( ˜ Fp)≥ord(F). Proposition 2.3. Let X∈b X2(C2,0). Let ˜ Xbe the formal vector field in (M, D)such that Dπ ·˜ X=X◦π. If pis a point of the tangent cone of Xthen ˜ Xp∈b X2(M, p). Proof: Let X=a(x, y)∂ ∂x +b(x, y)∂ ∂y with a(x, y) = ak(x, y) + ···, b(x, y) = bk(x, y) + ··· and k≥2. Let U1and U2be two charts of M=U1∪U2as in the proposition above. Then ˜ Xis given in the chart U1by ˜ X(x, v)=a(x, xv)∂ ∂x +b(x, xv)−va(x, xv) x ∂ ∂v =xk(ak(1, v)+x(···)) ∂ ∂x +xk−1((bk(1, v)−vak(1, v))+x(···)) ∂ ∂y . Now, if p= (0, v0)∈Dis such that bk(1, v0)−v0ak(1, v0) = 0, then νp(a(x, xv)) ≥kand νpb(x,xv)−va(x,xv) x≥kso ˜ Xp∈b X2(M, p). We say that the singular point pis strictly singular if any time we write X=fX′, then pis a singular point of X′. Note that in the above statement any strictly singular point of ˜ Xis in the tangent cone of X. Let us also recall that Seidenberg’s reduction of singularities [7] is done by blowing-up at strictly singular points. Lemma 2.4. Let F∈Diff1(C2,0) and X∈b X2(C2,0) such that F= Exp(X). Let ˜ Xbe as in the proposition above. Then for any p∈D ˜ Fp= Exp( ˜ Xp). Proof: Let U≃C2be a chart of Msuch that π|U:U→C2is defined by π(x, v) = (x, xv) and p∈U∩D={(0, v)∈U}be a point on the divisor. Without loss of generality, applying a linear change of coordinates, we can suppose that p= (0,0) ∈U. Since F(x, y) = Exp(X) = (exp X(x),exp X(y)),using the definition of ˜ F, we have ˜ F(x, v) = exp X(x),exp X(xv) exp X(x)= exp ˜ X(x),exp ˜ X(xv) exp ˜ X(x)! = exp ˜ X(x),exp ˜ X(x) exp ˜ X(v) exp ˜ X(x)!= (exp ˜ X(x),exp ˜ X(v)) = Exp( ˜ Xp)(x, v). Parabolic Curves for Diffeomorphisms in C2193 3. Existence of parabolic curves We need the following formal version of Camacho-Sad’s theorem [4] whose proof goes exactly as the original one (see also [5]). Theorem 3.1 (Camacho and Sad).Take X∈b X2(C2,0) with an isolated singularity at the origin. There is a desingularization morphism σ: ( ˜ M, ˜ D)→(C2,0) composition of a finite sequence of blow-ups with centers at strictly singular points and a point p∈˜ Dsatisfying the following property: There are local coordinates (u, v)at psuch that ˜ Dp= (u= 0) and the transform X∗of Xat pis of the form: X∗(u, v) = um(λu +u2(···)) ∂ ∂u + (µv +u(···)) ∂ ∂v  where λ6= 0,µ λ/∈Q>0and m≥ν(X)−1. Remark 3.2.The statement above is also valid when Xhas a dicritical desingularization; we just need to consider one of the infinitely many nondegenerate characteristic directions on a dicritical divisor. Let us prove Theorem 1.2. Take Xthe infinitesimal generator of F, and consider X∗and pas in Camacho-Sad’s Theorem. By Lemma 2.4 we have F∗ p(u, v) = Exp(X∗ p) = (u+λum+1 +O(um+2), v +µumv+O(um+1)) so F∗ pis a diffeomorphism tangent to the identity, with [1,0] as a nondegenerate characteristic direction. By Hakim’s Theorem, there exist ord(F∗ p)−1 disjoint parabolic curves ϕj: Ωj→˜ Mfor F∗ ptangent to the direction [1,0] at p. Since this direction is transversal to the divisor, it follows that ϕj(Ωj)∩˜ D={p}and thereby σ◦ϕjis also a parabolic curve for F. This ends the proof. Remark 3.3.In the case X=xkX′and S= (x= 0) invariant by X′, Camacho-Sad’s index of Xat 0 along Sis exactly Abate’s residual index of Fat 0 along S. Furthermore, according to J. Cano’s proof [5] of Camacho-Sad’s theorem, to find the points p∈˜ Dthat satisfy CamachoSad’s theorem, it is enough to follow after the first blow up, the singularities with Camacho-Sad’s index not in Q≥0. Thus, there exist parabolic curves for any characteristic direction of Fthat gives at the divisor Abate’s residual index not in Q≥0(see Corollary 3.1 in [1]). 194 F. E. Brochero Mart´ ınez, F. Cano, L. L´ opez-Hernanz References [1] M. Abate, The residual index and the dynamics of holomorphic maps tangent to the identity, Duke Math. J. 107(1) (2001), 173–207. [2] M. Abate, F. Bracci, and F. Tovena, Index theorems for holomorphic self-maps, Ann. of Math. (2) 159(2) (2004), 819–864. [3] C. A. Briot and J. C. Bouquet, Recherches sur les propri´et´es des fonctions d´efinies par des ´equations diff´erentielles, J. Ecole Polytechnique XXI (1856), 133–198. [4] C. Camacho and P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2) 115(3) (1982), 579–595. [5] J. Cano, Construction of invariant curves for singular holomorphic vector fields, Proc. Amer. Math. Soc. 125(9) (1997), 2649–2650. [6] M. Hakim, Analytic transformations of (Cp,0) tangent to the identity, Duke Math. J. 92(2) (1998), 403–428. [7] A. Seidenberg, Reduction of singularities of the differential equation A dy =B dx,Amer. J. Math. 90 (1968), 248–269. F. E. Brochero Mart´ınez: Departamento de Matem´atica UFMG Belo Horizonte, MG 30123-970 Brazil E-mail address:[email protected] F. Cano and L. L´opez-Hernanz: Departamento de ´ Algebra, Geometr´ıa y Topolog´ıa Universidad de Valladolid Spain E-mail address:[email protected] E-mail address:[email protected] Primera versi´o rebuda el 22 de febrer de 2007, darrera versi´o rebuda el 7 de setembre de 2007.