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

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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

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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
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: (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

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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

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