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

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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 :
37F80 - Higher-dimensional holomorphic and meromorphic dynamics
 37C25 - Fixed points, periodic points, fixed-point index theory
 32M25 - Complex vector fields
    Information on the Video

    Language : 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
Information on the Event

Event Title : Foliations, birational geometry and applications - Thematic Month Week 2 / Feuilletages, géométrie birationnelle et applications - Mois thématique semaine 2
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.20299303
Cite this video as: (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

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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