English
Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 2
Multi angle
Auteurs : López-Hernanz, Lorena (Auteur de la Conférence)
CIRM (Editeur )
37F80 - Higher-dimensional holomorphic and meromorphic dynamics
37C25 - Fixed points, periodic points, fixed-point index theory
32M25 - Complex vector fields
CIRM (Editeur )
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Résumé :
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.
Keywords : holomorphic dynamics; stable manifolds; invariant curves
Codes MSC : Keywords : holomorphic dynamics; stable manifolds; invariant curves
37F80 - Higher-dimensional holomorphic and meromorphic dynamics
37C25 - Fixed points, periodic points, fixed-point index theory
32M25 - Complex vector fields
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Informations sur la Vidéo
Réalisateur : Récanzone, LucaLangue : Anglais Date de publication : 03/03/2025 Date de captation : 04/02/2025 Sous collection : Research school arXiv category : Dynamical Systems ; Complex Variables Domaine : Dynamical Systems & ODE Format : MP4 (.mp4) - HD Durée : 00:59:29 Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2025-02-04_Lopez-Hernanz_2 
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Informations sur la Rencontre
Nom de la rencontre : Foliations, birational geometry and applications - Thematic Month Week 2 / Feuilletages, géométrie birationnelle et applications - Mois thématique semaine 2Organisateurs de la rencontre : Araujo, Carolina ; Belotto Da Silva, André ; Pichon, Anne ; Rond, Guillaume ; Ruggiero, Matteo Dates : 03/02/2025 - 07/02/2025 Année de la rencontre : 2025 URL Congrès : https://conferences.cirm-math.fr/3268.html
Données de citation
DOI : 10.24350/CIRM.V.20299203Citer cette vidéo: López-Hernanz, Lorena (2025). Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20299203 URI : http://dx.doi.org/10.24350/CIRM.V.20299203 |
Voir aussi
- [Multi angle] An introduction to the Minimal Model Program for foliations lecture 3 / Auteur de la Conférence Spicer, Calum.
- [Multi angle] An introduction to the Minimal Model Program for foliations lecture 2 / Auteur de la Conférence Spicer, Calum.
- [Multi angle] An introduction to the Minimal Model Program for foliations lecture 1 / Auteur de la Conférence Spicer, Calum.
- [Multi angle] Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 3 / Auteur de la Conférence López-Hernanz, Lorena.
- [Multi angle] Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 1 / Auteur de la Conférence López-Hernanz, Lorena.
- [Multi angle] Resolution of singularities for the dynamical mathematician lecture 3 / Auteur de la Conférence Abramovich, Dan.
- [Multi angle] Resolution of singularities for the dynamical mathematician lecture 2 / Auteur de la Conférence Abramovich, Dan.
- [Multi angle] Resolution of singularities for the dynamical mathematician lecture 1 / Auteur de la Conférence Abramovich, Dan.
Bibliographie
- 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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