Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T00:45:49.521Z Has data issue: false hasContentIssue false
  • English
  • Français

Summable formal invariant curves of diffeomorphisms

Published online by Cambridge University Press:  14 March 2011

L. LÓPEZ-HERNANZ
L. LÓPEZ-HERNANZ*
Affiliation:
Departamento de Álgebra, Geometría y Topología, Universidad de Valladolid, Spain (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let F be a tangent to the identity diffeomorphism in (ℂ2,0) and X its infinitesimal generator. We prove that Camacho and Sad’s formal invariant curves of X give summable formal power series, whose sums correspond to the parabolic curves found by Hakim for F and F−1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 211 - 221
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abate, M.. The residual index and the dynamics of holomorphic maps tangent to the identity. Duke Math. J. 107(1) (2001), 173–207.Google Scholar
[2]Briot, C. A. and Bouquet, J. C.. Recherches sur les propriétés des fonctions définies par des équations différentielles. J. École Polytech. XXI (1856), 133–198.Google Scholar
[3]Brochero Martínez, F. E., Cano, F. and López-Hernanz, L.. Parabolic curves for diffeomorphisms in . Publ. Mat. 52(1) (2008), 189–194.Google Scholar
[4]Brochero Martínez, F. E. and López-Hernanz, L.. Gevrey class of the infinitesimal generator of a diffeomorphism. Astérisque 323 (2009), 33–40.Google Scholar
[5]Camacho, C. and Sad, P.. Invariant varieties through singularities of holomorphic vector fields. Ann. of Math. (2) 115(3) (1982), 579–595.Google Scholar
[6]Canalis-Durand, M., Ramis, J. P., Schäfke, R. and Sibuya, Y.. Gevrey solutions of singularly perturbed differential equations. J. Reine Angew. Math. 518 (2000), 95–129.Google Scholar
[7]Cano, J.. An extension of the Newton–Puiseux polygon construction to give solutions of Pfaffian forms. Ann. Inst. Fourier (Grenoble) 43(1) (1993), 125–142.Google Scholar
[8]Écalle, J.. Les fonctions résurgentes. Tome III. L’équation du pont et la classification analytique des objects locaux. (Publications Mathématiques d’Orsay, 85-5). Université de Paris-Sud, Département de Mathématiques, Orsay, 1985.Google Scholar
[9]Fatou, P.. Sur les équations fonctionelles. Bull. Soc. Math. France 47 (1919), 161–271.CrossRefGoogle Scholar
[10]Fruchard, A. and Zhang, C.. Remarques sur les développements asymptotiques. Ann. Fac. Sci. Toulouse Math. (6) 8(1) (1999), 91–115.CrossRefGoogle Scholar
[11]Hakim, M.. Analytic transformations of (Cp,0) tangent to the identity. Duke Math. J. 92(2) (1998), 403–428.Google Scholar
[12]Ince, E. L.. Ordinary Differential Equations. Dover, New York, 1944.Google Scholar
[13]Leau, L.. Étude sur les équations fonctionelles à une ou à plusieurs variables. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 11 (1897), 1–110.Google Scholar
[14]Loray, F.. Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. Prépublication IRMAR, 2005.Google Scholar
[15]Milnor, J.. Dynamics in One Complex Variable (Annals of Mathematics Studies, 160). Princeton University Press, Princeton, NJ, 2006.Google Scholar
[16]Ramis, J. P.. Les séries k-sommables et leurs applications. Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory (Proc. Internat. Colloq., Centre for Physics, Les Houches, 1979) (Lecture Notes in Physics, 126). Springer, Berlin, 1980, pp. 178–199.Google Scholar
[17]Ribón, J.. Families of diffeomorphisms without periodic curves. Michigan Math. J. 53(2) (2005), 243–256.Google Scholar