Tips of Tongues in the Double Standard Family

Abstract

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 \[F_λ(x) := 2x + a+ \frac{b}π \sin(2πx){\quad\text{with}\quad} λ:=(a,b)\in \mathbb{R}/\mathbb{Z}\times (0,1).\] We prove that if $F_{\lambda }^{\circ n}-\mathrm{id}$ 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 $F_{\mu }^{\circ n}-\mathrm{id}$ 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.