# Tips of Tongues in the Double Standard Family

## TL;DR

This paper characterizes the local structure of parameter regions ('tips of tongues') where degree 2 circle maps have multiple zeros, showing they form regular cusps in parameter space.

## Contribution

It provides a detailed local analysis of bifurcation points in degree 2 circle maps, revealing the cusp structure of the tips of tongues.

## Key findings

- Tips of tongues are regular cusps in parameter space.
- Zero of multiplicity 3 implies a cusp in the bifurcation diagram.
- Local coordinates can be chosen to reveal the cusp structure.

## 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.