Boundaries of the Arnol'd tongues and the standard family

Abstract

For a family $(F_{t,a} : x \mapsto x + t + a\phi(x))$ of increasing homeomorphisms of $\mathbb R$ with $\phi$ 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 \textsf{Arnol'd tongue} $\mathcal T_\theta$ of \textsf{translation number} $\theta$ in the parameter space. Given a rational $p/q$, it is shown that the boundary $\partial \mathcal T_{p/q}$ 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 $\mathcal T_{p/q}$ have the same tangency at $a=0$ for $q\ge 2$ and it is known that $q$ is their \textsf{order of contact}. Using the techniques of \textsf{guided} and \textsf{admissible family}, we give a new proof of this. In particular we relate this to the \textsf{parabolic multiplicity} of the map $s_{p/q} : z \mapsto e^{i2\pi p/q}ze^{\pi z}$ at $0$.