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Meromorphic maps of finite type: parameter space
Multi angle
Authors : Fagella, Nuria (Author of the conference)
CIRM (Publisher )
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
CIRM (Publisher )
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Abstract :
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
MSC Codes : Keywords : meronorphic functions; Julia sets; J-stability; virtual centers; bifurcations
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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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 02/11/2021 Conference Date : 22/09/2021 Subseries : Research talks arXiv category : Dynamical Systems ; Distributed, Parallel, and Cluster Computing Mathematical Area(s) : Dynamical Systems & ODE Format : MP4 (.mp4) - HD Video Time : 00:59:41 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2021-09-22_Fagella.mp4 
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Information on the Event
Event Title : Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexeEvent Organizers : Benini, Anna Miriam ; Drach, Kostiantyn ; Dudko, Dzmitry ; Hlushchanka, Mikhail ; Schleicher, Dierk Dates : 20/09/2021 - 24/09/2021 Event Year : 2021 Event URL : https://conferences.cirm-math.fr/2546.html
Citation Data
DOI : 10.24350/CIRM.V.19812303Cite this video as: 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 |
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Bibliography
- 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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