CIRM - Videos & books Library - Box renormalization as a 'black box'

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Résumé : The concept of a complex box mapping (or puzzle mapping) is a generalization of the classical notion of polynomial-like map to the case when one allows for countably many components in the domain and finitely many components in the range of the mapping. In one-dimensional dynamics, box mappings appear naturally as first return maps to certain nice sets, and hence one arrives at a notion of box renormalization. We say that a rational map is box renormalizable if the first return map to a well-chosen neighborhood of the set of critical points (intersecting the Julia set) has a structure of a box mapping. In our talk, we will discuss various features of general box mappings, as well as so-called dynamically natural box mappings, focusing on their rigidity properties. We will then show how these results can be used almost as 'black boxes' to conclude similar rigidity properties for box renormalizable rational maps. We will give several examples to illustrate this procedure, these examples include, most prominently, complex polynomials of arbitrary degree and their Newton maps. (The talk is based on joint work with Trevor Clark, Oleg Kozlovski, Dierk Schleicher and Sebastian van Strien.)

Keywords : Box mapping; puzzle map; renormalization; rigidity; local connectivity

Codes MSC :
37F10 - Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F31 - Quasiconformal methods in holomorphic dynamics; quasiconformal dynamics
37F46 - Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets

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https://videos.cirm-math.fr/2021-09-23_Drach.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.19811903
Citer cette vidéo: (2021). Box renormalization as a 'black box'. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19811903
URI : http://dx.doi.org/10.24350/CIRM.V.19811903

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Bibliographie

CLARK, Trevor, DRACH, Kostiantyn, KOZLOVSKI, Oleg, et al. The dynamics of complex box mappings. arXiv preprint arXiv:2105.08654, 2021. - https://arxiv.org/abs/2105.08654

DRACH, Kostiantyn et SCHLEICHER, Dierk. Rigidity of Newton dynamics. arXiv preprint arXiv:1812.11919, 2018. - https://arxiv.org/abs/1812.11919v2

DRACH, Kostiantyn, LODGE, Russell, SCHLEICHER, Dierk, et al. Puzzles and the Fatou–Shishikura injection for rational Newton maps. Transactions of the American Mathematical Society, 2021, vol. 374, no 4, p. 2753-2784. - https://doi.org/10.1090/tran/8273

DRACH, Kostiantyn, MIKULICH, Yauhen, RÜCKERT, Johannes, et al. A combinatorial classification of postcritically fixed Newton maps. Ergodic Theory and Dynamical Systems, 2019, vol. 39, no 11, p. 2983-3014. - https://doi.org/10.1017/etds.2018.2

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