Singularities that are inaccessible by geometry. (Singularités non abordables par la géométrie.) (French) Zbl 0940.32013

Summary: This paper is devoted to local analytic objects (i.e. germs of vector fields or diffeomorphisms) in any dimension, with special emphasis on the interplay between the two main difficulties: small denominators and resonance. We introduce an arborification technique, which is well suited for tackling diophantine small denominators and we recall the definition of resurgent functions and monomials, which are essential in any resonant context. We show how a single equation, the so-called Bridge Equation, not only yields all holomorphic invariants (i.e. all analytic invariants depending holomorphically on the object) but also most intrinsic properties of local objects, such as: sectorial normalization, criteria for the existence of invariant analytic varieties, etc.


32S65 Singularities of holomorphic vector fields and foliations
Full Text: DOI Numdam EuDML


[1] [12] Foliations of the plane, Symposium on Differential Equations and Dynamical Systems, Warwick, 1968-1969, pp. 104-105. · Zbl 0272.34018
[2] [10] Formalisme lagrangien, Symposium on Differential Equations and Dynamical Systems, Warwick, 1968-1969, pp. 9-11.
[3] [E0] , Théorie des invariants holomorphes, Thèse, Orsay, 1974.
[4] [E1] , Les fonctions résurgentes, Vol. 1, Algèbres de fonctions résurgentes, Pub. Math. Orsay, (1981). · Zbl 0499.30034
[5] [E2] , Les fonctions résurgentes, Vol. 2, Les fonctions résurgentes appliquées à l’itération, Pub. Math. Orsay, (1981). · Zbl 0499.30035
[6] [E3] , Les fonctions résurgentes, Vol. 3, L’équation du pont et la classification analytique des objets locaux, Pub. Math. Orsay, (1985). · Zbl 0602.30029
[7] [E4] , Classification analytique des champs de vecteurs locaux résonnants de Cv. Actes de la 37ème rencontre entre Mathématiciens et Physiciens (RCP), Strasbourg, Oct. 1983.
[8] [E5] , Finitude des cycles-limites et accéléro-sommation de l’application de retour (pp. 74-159) in : Bifurcations of Planar Vector fields ; Proc. Luminy 1989, Lect. Notes 1455, Springer. · Zbl 0729.34016
[9] [E6] , Introduction aux fonctions analysables et application à la preuve constructive de la conjecture de Dulac, Hermann, Paris, 1991. · Zbl 1241.34003
[10] [E7] , Calcul accélératoire et applications (à paraître aux Actualités Math., Ed. Hermann, Paris).
[11] [E8] , Calcul compensatoire et linéarisation quasianalytique des objets locaux. A paraître aux Proc. of the Coll. on complex Anal. Methods in Dyn. Syst., IMPA, Rio de Janeiro, Jan. 1992.
[12] [21] Invariants of foliations, Global Analysis and Its Applications, International Centre for Theoretical Physics, Trieste, 1972, vol. II, 215-219. · Zbl 0741.30030
[13] [E11] , Cohesive functions and weak accelerations. A paraître au Journal d’Analyse Math., Szolem Mandelbrojt Memorial Volume, 1992. · Zbl 0808.30002
[14] [MR1] et , Problèmes de modules pour des équations différentielles non-linéaires du premier ordre, Pub. Math. IHES, 55 (1982), 63-16. · Zbl 0546.58038
[15] [MR2] et , Classification analytique des équations différentielles non-linéaires résonnantes du premier ordre, Ann. Sc. Ec. Norm. Sup., 4ème série, t.16 (1983), 571-625. · Zbl 0534.34011
[16] [MR3] et , Elementary acceleration and multisummability, Paris, 1990. · Zbl 0748.12005
[17] [Ma] , Normalisation des champs de vecteurs holomorphes, d’après A.D. Brjuno, Séminaire Bourbaki, 1980-1981, n° 564-01. · Zbl 0481.34013
[18] [Rü1] , Über die Iteration analytischer Funktionen, J. Math. Mech., 17, 523-532. · Zbl 0186.47704
[19] [Rü2] , On the convergence of power series transformations of analytic mappings near a fixed point into a normal form, Preprint IHES, Paris, 1977.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.