Nonlinear Analysis: Theory, Methods & Applications
Equivalence and semi-completude of foliations☆
Introduction
Let X be a holomorphic vector field in with an isolated singularity at the origin and such that the eigenvalues of its linear part are both non zero. Then the foliation associated to X admits a semi-complete representative [11]. We can ask if this result is still valid for higher dimensions.
In , the Siegel Domain is a thin set (it has zero measure). Contrary to the case, the interior of the Siegel Domain is non empty for , when ; thus the Siegel Domain represents an important set for the problem above. As the conclusions for the Poincaré Domain are easy, we will focus on the Siegel Domain.
Let , , be a holomorphic vector field, where represent the eigenvalues of , verifying:
- (a)
the origin is an isolated singularity,
- (b)
X is of Siegel type (0 belongs to the convex hull of ),
- (c)
all eigenvalues of are non zero and there exists a straight line through the origin, in the complex plane, separating from the other eigenvalues,
- (d)
up to a change of coordinates, , where and for all i.
Up to multiplication by a constant we can assume that .
In this paper only vector fields with an isolated singularity at the origin are considered, even if not explicitly stated.
It is important to remark that if and X is a vector field with an isolated singularity at the origin and of strict Siegel type (the convex hull of contains a neighbourhood of the origin), then (c) and (d) are immediately satisfied [1]; in particular, there exists at least one eigenvalue such that the angle between and the other eigenvalues is greater than (eigenvalues viewed as vectors in ).
Let X be a holomorphic vector field on , with an isolated singularity at the origin and of “strict Siegel type” (, where and are the eigenvalues of ). Then X admits two separatrices. Let be a transversal section to one of the separatrices and the foliation associated to X. It is well known that the saturated of by together with the other separatrix contains a neighbourhood of the origin [9]. For higher dimension we obtain: Proposition Let X be a holomorphic vector field verifying (a), (b), (c) and (d) and S the separatrix tangent to the eigenspace associated to . The saturated of a transversal section to S, at a point sufficiently close to the origin, together with the invariant manifold transverse to that separatrix contains a neighbourhood of the origin.
The method used in the proof of the proposition allows us to prove: Theorem Let be the foliation associated to a holomorphic vector field X verifying (a), (b), (c) and (d). Then admits a semi-complete vector field as its representative, in a neighbourhood of the singularity.
In particular, any vector field on of strict Siegel type admits a semi-complete representative.
It is obvious that if X and Y, two holomorphic vector fields, are analytically equivalent in a neighbourhood of a singularity then the holonomies relatively to the separatrices, if they exist, are analytically conjugated (there exist holomorphic vector fields, defined on a neighbourhood of the origin of , , without separatrices [5], [7]).
Mattei and Moussu proved the reciprocal of this result for linearization of vector fields on of strict Siegel type (if the eigenvalues of the linear part, at the singularity, are in the Poincaré Domain and do not verify any resonance relation between them, then both X and the holonomy are linearizable) and, later, Mattei proved that if the holonomies are analytically conjugated then the vector fields are analytically equivalent [8].
An extension, in a sense, of the results of Mattei and Moussu is here obtained: Theorem Let X and Y be two vector fields verifying (a), (b), (c) and (d). Denote by and the holonomies of X and Y relatively to the separatrices of X and Y tangent to the eigenspace associated to the first eigenvalue, respectively. Then if and are analytically conjugated, X and Y are analytically equivalent.
Section snippets
Vector fields of Siegel type
Let and be holomorphic vector fields with a singularity at the origin. We say that X is analytically conjugated to Y in a neighbourhood of the origin if there exists a holomorphic diffeomorphism , where , , such that and We say that X and Y are analytically equivalent if X is analytically conjugated to fY, for some holomorphic function f verifying .
The integral curves of any vector field X define a foliation of complex dimension
Semi-completude of vector fields of strict Siegel type
The definition of a semi-complete vector field relatively to a (relatively compact) open set U was introduced in [10]. Definition 1 Let X be a holomorphic vector field defined on a complex manifold M and an open subset of M. We say that X is semi-complete relatively to U if there exists a holomorphic application where is an open set containing such that , , , when the two members are defined, and .
We call
Acknowledgements
I would like to thank Professor J. Basto-Gonçalves for the discussions on the subject.
References (11)
- et al.
The topology of holomorphic flows with singularity
Publ. Math. Inst. Hautes Étud. Sci.
(1978) Holomorphic flows in with resonances
Trans. Amer. Math. Soc.
(1992)- et al.
Remarks in the orbital analytic classification of germs of vector fields
Math. USSR Sb.
(1984) - et al.
Singularités des flots holomorphes II
Ann. Inst. Fourier
(1997) - et al.
Germs of holomorphic vector fields in without a separatrix
Invent. Math.
(1992)
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Financial support from Fundação para a Ciência e Tecnologia (FCT) through Centro de Matemática da Universidade do Porto, from PRODEPIII and Fundação Calouste Gulbenkian.