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Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 3
Multi angle
Authors : López-Hernanz, Lorena (Author of the conference)
CIRM (Publisher )
37F80 - Higher-dimensional holomorphic and meromorphic dynamics
37C25 - Fixed points, periodic points, fixed-point index theory
32M25 - Complex vector fields
CIRM (Publisher )
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Abstract :
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
MSC Codes : 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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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 03/03/2025 Conference Date : 06/02/2025 Subseries : Research school arXiv category : Dynamical Systems ; Complex Variables Mathematical Area(s) : Dynamical Systems & ODE Format : MP4 (.mp4) - HD Video Time : 00:57:00 Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2025-02-06_Lopez-Hernanz_3 
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Information on the Event
Event Title : Foliations, birational geometry and applications - Thematic Month Week 2 / Feuilletages, géométrie birationnelle et applications - Mois thématique semaine 2Event Organizers : Araujo, Carolina ; Belotto Da Silva, André ; Pichon, Anne ; Rond, Guillaume ; Ruggiero, Matteo Dates : 03/02/2025 - 07/02/2025 Event Year : 2025 Event URL : https://conferences.cirm-math.fr/3268.html
Citation Data
DOI : 10.24350/CIRM.V.20299303Cite this video as: López-Hernanz, Lorena (2025). Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 3. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20299303 URI : http://dx.doi.org/10.24350/CIRM.V.20299303 |
See Also
- [Multi angle] An introduction to the Minimal Model Program for foliations lecture 3 / Author of the conference Spicer, Calum.
- [Multi angle] An introduction to the Minimal Model Program for foliations lecture 2 / Author of the conference Spicer, Calum.
- [Multi angle] An introduction to the Minimal Model Program for foliations lecture 1 / Author of the conference Spicer, Calum.
- [Multi angle] Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 2 / Author of the conference López-Hernanz, Lorena.
- [Multi angle] Local dynamics of tangent to the identity biholomorphisms in dimension two lecture 1 / Author of the conference López-Hernanz, Lorena.
- [Multi angle] Resolution of singularities for the dynamical mathematician lecture 3 / Author of the conference Abramovich, Dan.
- [Multi angle] Resolution of singularities for the dynamical mathematician lecture 2 / Author of the conference Abramovich, Dan.
- [Multi angle] Resolution of singularities for the dynamical mathematician lecture 1 / Author of the conference Abramovich, Dan.
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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