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 \[F_\lambda(x) := 2x + a+ \frac{b}{\pi} \sin(2\pi x){\quad\text{with}\quad} \lambda:=(a,b)\in \mathbb{R}/\mathbb{Z}\times (0,1).\] 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.