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Tips of tongues in the double standard family - Buff, Xavier (Author of the conference) | CIRM H

Virtualconference

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.[-]
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 ...[+]

37F10 ; 37F45 ; 37G10

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