Foliations, birational geometry and applications - Thematic Month Week 2 / Feuilletages, géométrie birationnelle et applications - Mois thématique semaine 2

Collection Foliations, birational geometry and applications - Thematic Month Week 2 / Feuilletages, géométrie birationnelle et applications - Mois thématique semaine 2

Organizer(s) Araujo, Carolina ; Belotto Da Silva, André ; Pichon, Anne ; Rond, Guillaume ; Ruggiero, Matteo

Date(s) 03/02/2025 - 07/02/2025

linked URL https://conferences.cirm-math.fr/3268.html

00:00:00 / 00:00:00

7 9

Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 3

By Lorena López-Hernanz

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.

Information about the video

Date of recording 06/02/2025
Date of publication 03/03/2025
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.20299303
Cite this video López-Hernanz, Lorena (06/02/2025). Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 3. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20299303
URL https://dx.doi.org/10.24350/CIRM.V.20299303

Domain(s)

Bibliography

ABATE, Marco. The residual index and the dynamics of holomorphic maps tangent to the identity. Duke Math. J. 107 (2001), no. 1, 173-207 - https://doi.org/10.1215/S0012-7094-01-10719-9
ECALLE, Jean. Les fonctions résurgentes. Tome III. L'équation du pont et la classification analytique des objects locaux. Publications Mathématiques d'Orsay, 85-5, 1985 - http://sites.mathdoc.fr/PMO/PDF/E_ECALLE_85_05.pdf
HAKIM, Monique. Analytic transformations of (C p, 0) tangent to the identity. Duke Math. J. 92 (1998) n°2, 403-428. - https://doi.org/10.1215/S0012-7094-98-09212-2
LÓPEZ-HERNANZ, Lorena et SANZ SÁNCHEZ, Fernando. Parabolic curves of diffeomorphisms asymptotic to formal invariant curves. Journal für die reine und angewandte Mathematik (Crelles Journal), 2018, vol. 2018, no 739, p. 277-296. - https://doi.org/10.1515/crelle-2015-0064
LÓPEZ-HERNANZ, Lorena, RIBÓN, Javier, SÁNCHEZ, Fernando Sanz, et al. Stable manifolds of biholomorphisms in $\mathbb {C}^ n $ asymptotic to formal curves. Proc. Lond. Math. Soc. 125 (2020), no. 2, 277-317. - https://doi.org/10.1112/plms.12447
LÓPEZ-HERNANZ, Lorena et ROSAS, Rudy. Characteristic directions of two-dimensional biholomorphisms. Compositio Mathematica, 2020, vol. 156, no 5, p. 869-880. - https://doi.org/10.1112/S0010437X20007071
LÓPEZ-HERNANZ, Lorena, RAISSY, Jasmin, RIBÓN, Javier, et al. Stable manifolds of two-dimensional biholomorphisms asymptotic to formal curves. International Mathematics Research Notices, 2021, vol. 2021, no 17, p. 12847-12887. - https://doi.org/10.1093/imrn/rnz143

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