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}$\to $\mathbb{R}$/$\mathbb{Z}$$ defined by $$F_{\lambda }(x):=2x+a+\frac{b}{\pi }\sin (2\pi x)\text{with}\lambda :=(a,b)\in \mathbb{R}\mathrm{/}\mathbb{Z}\times (0,1)\mathrm{.}$$ We prove that if $F_\lambda^{\circ n}-{\rm id}$hasazeroofmultiplicity$3$in$$\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 $F_{\mu }^{\circ n}-\mathrm{i}\mathrm{d}$ 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.