Français
Tips of tongues in the double standard family
Virtualconference
Authors : Buff, Xavier (Author of the conference)
CIRM (Publisher )
37F10 - Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 37G10 - Bifurcations of singular points
CIRM (Publisher )
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Abstract :
We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree 2 circle maps $F_{\lambda}: \mathbb{R} / \mathbb{Z} \rightarrow \mathbb{R} / \mathbb{Z}$ defined by
$$
F_{\lambda}(x):=2 x+a+\frac{b}{\pi} \sin (2 \pi x), \text { with } \lambda:=(a, b) \in \mathbb{R} / \mathbb{Z} \times(0,1)
$$
We prove that if $F_{\lambda o}^{\circ n}-$ id has a zero of multiplicity 3 in $\mathbb{R} / \mathbb{Z}$, then there is a system of local coordinates $(\alpha, \beta): W \rightarrow \mathbb{R}^{2}$ defined in a neighborhood $W$ of $\lambda_{0}$, such that $\alpha\left(\lambda_{0}\right)=\beta\left(\lambda_{0}\right)=0$ and $F_{\lambda}^{\circ n}-$ id has a multiple zero with $\lambda \in W$ if and only if $\beta^{3}(\lambda)=\alpha^{2}(\lambda)$. This shows that the tips of tongues are regular cusps. This is joint work with K. Banerjee, J. Canela and A. Epstein.
Keywords : Holomorphic dynamics; transcendental dynamics; transversality; quadratic differentials
MSC Codes : $$
F_{\lambda}(x):=2 x+a+\frac{b}{\pi} \sin (2 \pi x), \text { with } \lambda:=(a, b) \in \mathbb{R} / \mathbb{Z} \times(0,1)
$$
We prove that if $F_{\lambda o}^{\circ n}-$ id has a zero of multiplicity 3 in $\mathbb{R} / \mathbb{Z}$, then there is a system of local coordinates $(\alpha, \beta): W \rightarrow \mathbb{R}^{2}$ defined in a neighborhood $W$ of $\lambda_{0}$, such that $\alpha\left(\lambda_{0}\right)=\beta\left(\lambda_{0}\right)=0$ and $F_{\lambda}^{\circ n}-$ id has a multiple zero with $\lambda \in W$ if and only if $\beta^{3}(\lambda)=\alpha^{2}(\lambda)$. This shows that the tips of tongues are regular cusps. This is joint work with K. Banerjee, J. Canela and A. Epstein.
Keywords : Holomorphic dynamics; transcendental dynamics; transversality; quadratic differentials
37F10 - Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 37G10 - Bifurcations of singular points
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 02/11/2021 Conference Date : 24/09/2021 Subseries : Research talks arXiv category : Dynamical Systems Mathematical Area(s) : Dynamical Systems & ODE Format : MP4 (.mp4) - HD Video Time : 01:05:00 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2021-09-24_Buff.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.19811203Cite this video as: Buff, Xavier (2021). Tips of tongues in the double standard family. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19811203 URI : http://dx.doi.org/10.24350/CIRM.V.19811203 |
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Bibliography
- BANERJEE, Kuntal, BUFF, Xavier, CANELA, Jordi, et al. Tips of Tongues in the Double Standard Family. arXiv preprint arXiv:1903.01795, 2019. - https://arxiv.org/abs/1903.01795
- DEZOTTI, Alexandre. Connectedness of the Arnold tongues for double standard maps. Proceedings of the American Mathematical Society, 2010, p. 3569-3583. - https://doi.org/10.1090/S0002-9939-10-10355-4
- EPSTEIN A. Towers of finite type complex analytic maps, Ph. D. Thesis, CUNY, 1993. -
- MISIUREWICZ, Michał et RODRIGUES, Ana. Double standard maps. Communications in mathematical physics, 2007, vol. 273, no 1, p. 37-65. - https://doi.org/10.1007/s00220-007-0223-5
- MISIUREWICZ, Michał et RODRIGUES, Ana. On the tip of the tongue. Journal of Fixed Point Theory and Applications, 2008, vol. 3, no 1, p. 131-141. - https://doi.org/10.1007/s11784-008-0052-y
- MISIUREWICZ, Michał et RODRIGUES, Ana. Non-generic cusps. Transactions of the American Mathematical Society, 2011, vol. 363, no 7, p. 3553-3572. - https://doi.org/10.1090/S0002-9947-2011-05114-7
