CIRM - Videos & books Library - Transcendental dynamics and infinite-dimensional Thurston theory

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Authors : Bogdanov, Konstantin (Author of the conference)
CIRM (Publisher )

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Abstract : In complex dynamics it is usually important to understand the dynamical behavior of critical (or singular) orbits. For quadratic polynomials, this leads to the study of the Mandelbrot set and of its complement. In our talk we present a classification of some explicit families of the transcendental entire functions for which all singular values escape, i.e. functions belonging to the complement of the 'transcendental analogue' of the Mandelbrot set. This classification allows us to introduce higher dimensional analogues of parameter rays and to explore their properties. A key ingredient is a generalization of the famous Thurston's Topological Characterization of Rational Functions, but for the case of infinite rather than finite postsingular set. Analogously to Thurston's theorem, we consider the sigma-iteration on the Teichmüller space and investigate its convergence. Unlike the classical case, the underlying Teichmüller space is infinite-dimensional which leads to a completely different theory.

Keywords : Thurston; Teichmüller space; transcendental entire function; iteration; complex dynamics

MSC Codes :
37F20 - Combinatorics and topology in relation with holomorphic dynamical systems
37F34 - Teichmüller theory; moduli spaces of holomorphic dynamical systems

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2021-09-23_Bogdanov.mp4

Information on the Event

Event Title : Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe
Event 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.19810803
Cite this video as: Bogdanov, Konstantin (2021). Transcendental dynamics and infinite-dimensional Thurston theory. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19810803
URI : http://dx.doi.org/10.24350/CIRM.V.19810803

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Bibliography

BOGDANOV, Konstantin. Infinite-dimensional Thurston theory and transcendental dynamics I: infinite-legged spiders. arXiv preprint arXiv:2102.00300, 2021. - https://arxiv.org/abs/2102.00300

BOGDANOV, Konstantin. Infinite-dimensional Thurston theory and transcendental dynamics II: classification of entire functions with escaping singular orbits. arXiv preprint arXiv:2102.10728, 2021. - https://arxiv.org/abs/2102.10728

BOGDANOV, Konstantin. Infinite-dimensional Thurston theory and transcendental dynamics IV: dependence on parameters and escape on (pre-) periodic rays. arXiv preprint arXiv:2104.13102, 2021. - https://arxiv.org/abs/2104.13102

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