Boundaries of the Arnol'd tongues and the standard family

For a family $(F_{t,a}:x\mapsto x+t+a\varphi (x))$ of increasing homeomorphisms of $\mathbb{R}$ with $\varphi$ being Lipschitz continuous of period 1, there is a parameter space consisting of the values $(t,a)$ such that the map $F_{t,a}$ is strictly increasing and it induces an orientation preserving circle homeomorphism. For each $\theta \in \mathbb{R}$ there is an $\mathsf{\text{Arnol’d}}\mathsf{\text{tongue}}$ ${\mathcal{T}}_{\theta }$ of $\mathsf{\text{translation}}\mathsf{\text{number}}$ $\theta$ in the parameter space. Given a rational $p\mathrm{/}q$, it is shown that the boundary $\mathrm{\partial }{\mathcal{T}}_{p\mathrm{/}q}$ is a union of two Lipschitz curves which intersect at $a=0$ and there can be a non zero angle between them. In this direction we compute the first order asymptotic expansion of the boundaries of the rational and irrational tongues in the parameter space around $a=0$. For the standard family $(S_{t,a}:x\mapsto x+t+a\sin (2\pi x))$, the boundary curves of ${\mathcal{T}}_{p\mathrm{/}q}$ have the same tangency at $a=0$ for $q\ge 2$anditisknownthat$q$ is their $\mathsf{\text{order}}\mathsf{\text{of}}\mathsf{\text{contact}}$. Using the techniques of $\mathsf{\text{guided}}$ and $\mathsf{\text{admissible}}\mathsf{\text{family}}$, we give a new proof of this. In particular we relate this to the \textsf{parabolic multiplicity} of the map $s_{p\mathrm{/}q}:z\mapsto e^{i2\pi p\mathrm{/}q}z{e}^{\pi z}$ at $0$.