Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe

Collection Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe

Organizer(s) Benini, Anna Miriam ; Drach, Kostiantyn ; Dudko, Dzmitry ; Hlushchanka, Mikhail ; Schleicher, Dierk

Date(s) 20/09/2021 - 24/09/2021

linked URL https://conferences.cirm-math.fr/2546.html

00:00:00 / 00:00:00

25 27

Tips of tongues in the double standard family

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.

Information about the video

Date of recording 24/09/2021
Date of publication 02/11/2021
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19811203
Cite this video Buff, Xavier (24/09/2021). Tips of tongues in the double standard family. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19811203
URL https://dx.doi.org/10.24350/CIRM.V.19811203

Domain(s)

Dynamical Systems

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

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