×

Extending to the complex line Dulac’s corner maps of non-degenerate planar singularities. (English) Zbl 1328.30016

Summary: We study the complex Dulac map for a holomorphic foliation of the complex plane, near a non-degenerate singularity (both eigenvalues of the linearization are nonzero) with two separatrices. Following the well-known results of Il’yashenko we provide a geometric approach allowing to study the whole maximal domain of (geometric) definition of the Dulac map. In particular its topology and the regularity of its boundary are completely described. We also study the order of magnitude of the first non-trivial term of its asymptotic expansion and show how to compute it using path integrals supported in the leaves of the linearized foliation. Explicit bounds on the remainder are given. We perform similarly the study of the Dulac time spent around the singularity. All results are formulated in a unified framework taking no heed to the usual dynamical discrimination (i.e. no matter whether the singularity is formally orbitally linearizable or not and regardless of the arithmetic of the eigenvalues ratio).

MSC:

30E99 Miscellaneous topics of analysis in the complex plane
37F75 Dynamical aspects of holomorphic foliations and vector fields
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
32S65 Singularities of holomorphic vector fields and foliations
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Brjuno A D 1971 Analytic form of differential equations. I, II. Trans. Moscow Math. Soc.25 131–288
[2] Brjuno A D 1972 Analytic form of differential equations. I, II. Trans. Moscow Math. Soc.26 131–288
[3] Brjuno A D 1974 Analytic form of differential equations. I, II. Trans. Moscow Math. Soc.25 199–239
[4] Dulac H 1909 Sur les points singuliers d’une équation différentielle Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys.1 329–79 · JFM 41.0342.01 · doi:10.5802/afst.262
[5] Dulac H 1923 Sur les cycles limites Bull. Soc. Math. France51 45–188 · JFM 49.0304.01
[6] Écalle J 1992 Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac(Actualités Mathématiques (Current Mathematical Topics)) (Paris: Hermann)
[7] Il’yashenko Y S 1984 Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane Funktsional. Anal. i Prilozhen.18 199–209 · Zbl 0564.34034 · doi:10.1007/BF01086157
[8] Il’yashenko Y S 1985 Dulac’s memoir ’On limit cycles’ and related questions of the local theory of differential equations Usp. Mat. Nauk40 41–78
[9] Il’yashenko Y S 1985 Dulac’s memoir ’On limit cycles’ and related questions of the local theory of differential equations Usp. Mat. Nauk40 199
[10] Il’yashenko Y S 1991 Finiteness Theorems for Limit Cycles(Translations of Mathematical Monographs vol 94) (Providence, RI: American Mathematical Society) (translated from the Russian by H H McFaden)
[11] Ilyashenko Y and Yakovenko S 2008 Lectures on Analytic Differential Equations(Graduate Studies in Mathematics vol 86) (Providence, RI: American Mathematical Society) · Zbl 1186.34001
[12] Loray F 2010 Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux preprint
[13] Mourtada A and Moussu R 1997 Applications de Dulac et applications pfaffiennes Bull. Soc. Math. France125 1–13 · Zbl 0884.58004
[14] Marín D and Mattei J F 2008 Incompressibilité des feuilles de germes de feuilletages holomorphes singuliers Ann. Sci. Éc. Norm. Supér.41 855–903
[15] Marín D and Mattei J F 2014 Topology of singular holomorphic foliations along a compact divisor J. Singul.9 122–50 · Zbl 1304.32020
[16] Mardešić P, Marín D and Villadelprat J 2008 Unfolding of resonant saddles and the Dulac time Discrete Contin. Dyn. Syst.21 1221–44 · Zbl 1153.37388 · doi:10.3934/dcds.2008.21.1221
[17] Mardešić P and Saavedra M 2007 Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles Proc. Am. Math. Soc.135 3273–82 · Zbl 1127.34017 · doi:10.1090/S0002-9939-07-09026-0
[18] Perez Marco R 1992 Solution complète au problème de Siegel de linéarisation d’une application holomorphe au voisinage d’un point fixe (d’après J-C Yoccoz) Astérisque Exp. No. 753, 4, 273–310 Séminaire Bourbaki, vol 1991/92
[19] Seidenberg A 1968 Reduction of singularities of the differential equation Am. J. Math.90 248–69 · Zbl 0159.33303 · doi:10.2307/2373435
[20] Teyssier L 2015 Germes de feuilletages présentables du plan complexe Bull. Braz. Math. Soc.46 275–329 · Zbl 1337.32045
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.