CIRM - Videos & books Library - Meromorphic maps of finite type: parameter space

Multi angle

Auteurs : Fagella, Nuria (Auteur de la Conférence)
CIRM (Editeur )

Loading the player...

Résumé : In this talk we present bifurcation phenomena in natural families of rational or (transcendental) meromorphic functions of finite type $\left\{f_{\lambda}:=\varphi_{\lambda} \circ f_{\lambda_{0}} \circ \psi_{\lambda}^{-1}\right\}_{\lambda \in M}$, where $M$ is a complex connected manifold, $\lambda_{0} \in M, f_{\lambda_{0}}$ is a meromorphic map and $\varphi_{\lambda}$ and $\psi_{\lambda}$ are families of quasiconformal homeomorphisms depending holomorphically on $\lambda$ and with $\psi_{\lambda}(\infty)=\infty$. There are fundamental differences compared to the rational or entire setting due to the presence of poles and therefore of parameters for which singular values are eventually mapped to infinity (singular parameters). Under mild conditions we show that singular (asymptotic) parameters are the endpoint of a curve of parameters for which an attracting cycle progressively exits the domain, while its multiplier tends to zero, proving a conjecture from [Fagella, Keen, 2019]. We also present the connections between cycles exiting the domain, singular parameters, activity of singular orbits and $\mathcal{J}$-unstability, converging to a theorem in the spirit of Mañé-Sad-Sullivan's celebrated result.

Keywords : meronorphic functions; Julia sets; J-stability; virtual centers; bifurcations

Codes MSC :
30D05 - Functional equations in the complex domain, iteration and composition of analytic functions
30D30 - Meromorphic functions, general theory
37F10 - Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F46 - Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets

Informations sur la Vidéo

Réalisateur :

Hennenfent, Guillaume

Langue :

Date de publication :

Date de captation :

Sous collection :

arXiv category :

Domaine :

Format :

Durée :

Audience :

Download :

https://videos.cirm-math.fr/2021-09-22_Fagella.mp4

Informations sur la Rencontre

Nom de la rencontre : Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe
Organisateurs de la rencontre : Benini, Anna Miriam ; Drach, Kostiantyn ; Dudko, Dzmitry ; Hlushchanka, Mikhail ; Schleicher, Dierk
Dates : 20/09/2021 - 24/09/2021
Année de la rencontre : 2021
URL Congrès : https://conferences.cirm-math.fr/2546.html

Données de citation

DOI : 10.24350/CIRM.V.19812303
Citer cette vidéo: Fagella, Nuria (2021). Meromorphic maps of finite type: parameter space. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19812303
URI : http://dx.doi.org/10.24350/CIRM.V.19812303

Voir aussi

[Virtualconference] Renormalization in complex dynamics / Auteur de la Conférence Shishikura, Mitsuhiro.
[Multi angle] Characterizing Thurston maps by lifting trees / Auteur de la Conférence Winarski, Rebecca.
[Multi angle] Docile transcendental entire functions / Auteur de la Conférence Waterman, James.
[Virtualconference] Characterizing rational maps positively using graphs / Auteur de la Conférence Thurston, Dylan.
[Multi angle] Transcendental Thurston Theory for entire functions and compositions / Auteur de la Conférence Shemyakov, Sergey.
[Multi angle] Iterated monodromy groups and transcendental dynamics / Auteur de la Conférence Reinke, Bernhard.
[Multi angle] A dynamical homeomorphism between the Mandelbrot set $M$ and the parabolic Mandelbrot set $M_{1}$ / Auteur de la Conférence Petersen, Carsten Lunde.
[Virtualconference] Entire functions with Cantor bouquet Julia sets / Auteur de la Conférence Pardo-Simon, Leticia.
[Multi angle] Contracting self-similar groups and conformal dimension / Auteur de la Conférence Nekrashevych, Volodymyr.
[Virtualconference] Interbreeding in conformal dynamics, and its applications near and far / Auteur de la Conférence Mukherjee, Sabyasachi.
[Multi angle] Wandering lakes of Wada / Auteur de la Conférence Martí-Pete, David.
[Multi angle] Story of the Feigenbaum point / Auteur de la Conférence Lyubich, Mikhail.
[Multi angle] Transcendental Julia sets of minimal Hausdorff dimension / Auteur de la Conférence Lazebnik, Kirill.
[Multi angle] Introduction to Thurston's theorems / Auteur de la Conférence Hubbard, John H..
[Multi angle] Decomposition results in rational dynamics / Auteur de la Conférence Hlushchanka, Mikhail.
[Multi angle] Hyperbolicity of renormalization for bi-cubic circle maps with bounded combinatorics / Auteur de la Conférence Estevez Jacinto, Gabriela Alexandra.
[Virtualconference] Near-degenerate regime in neutral renormalization / Auteur de la Conférence Dudko, Dzmitry.
[Multi angle] Box renormalization as a 'black box' / Auteur de la Conférence Drach, Kostiantyn.
[Virtualconference] Arithmetic geometric models for the renormalisation of irrationally indifferent attractors / Auteur de la Conférence Cheraghi, Davoud.
[Virtualconference] Tips of tongues in the double standard family / Auteur de la Conférence Buff, Xavier.
[Multi angle] Eremenko's conjecture, Devaney's hairs, and the growth of counterexamples / Auteur de la Conférence Brown, Andrew.
[Virtualconference] The visual sphere of an expanding Thurston map / Auteur de la Conférence Bonk, Mario.
[Multi angle] Transcendental dynamics and infinite-dimensional Thurston theory / Auteur de la Conférence Bogdanov, Konstantin.
[Virtualconference] Transcendental functions with small singular sets / Auteur de la Conférence Bishop, Christopher.
[Multi angle] Polynomial versus transcendental dynamics / Auteur de la Conférence Benini, Anna Miriam.
[Multi angle] Quadratic polynomials / Auteur de la Conférence Bartholdi, Laurent.

Bibliographie

ASTORG, Matthieu, BENINI, Anna Miriam, et FAGELLA, Núria. Bifurcation loci of families of finite type meromorphic maps. arXiv preprint arXiv:2107.02663, 2021. - https://arxiv.org/abs/2107.02663

Fagella.png

Sélection Signaler une erreur

TUTELLES

PARTENAIRES

Destination de la recherche

Raccourcis

Voir aussi