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Article Dans Une Revue Nonlinearity Année : 2021

Tips of Tongues in the Double Standard Family

Kuntal Banerjee
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Xavier Buff
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Jordi Canela
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Résumé

We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree 2 circle maps $F_\lambda:\mathbb{R}/\mathbb{Z}\to \mathbb{R}/\mathbb{Z}$ defined by \[F_\lambda(x) := 2x + a+ \frac{b}{\pi} \sin(2\pi x){\quad\text{with}\quad} \lambda:=(a,b)\in \mathbb{R}/\mathbb{Z}\times (0,1).\] We prove that if $F_\lambda^{\circ n}-{\rm id}$ has a zero of multiplicity $3$ in $\mathbb{R}/\mathbb{Z}$, then there is a system of local coordinates $(\alpha,\beta):W\to \mathbb{R}^2$ defined in a neighborhood $W$ of $\lambda$, such that $\alpha(\lambda) =\beta(\lambda)=0$ and $F_\mu^{\circ n} - {\rm id}$ has a multiple zero with $\mu\in W$ if and only if $\beta^3(\mu) = \alpha^2(\mu)$. This shows that the tips of tongues are regular cusps.

Dates et versions

hal-02058582 , version 1 (06-03-2019)

Identifiants

Citer

Kuntal Banerjee, Xavier Buff, Jordi Canela, Adam Epstein. Tips of Tongues in the Double Standard Family. Nonlinearity, 2021, 34 (12), pp.8174-8191. ⟨hal-02058582⟩
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