On the widths of the Arnol'd Tongues

Abstract

Let $F: \mathbb R \to \mathbb R$ be a real analytic increasing diffeomorphism with $F-{\rm Id}$ being 1 periodic. Consider the translated family of maps $(F_t :\mathbb R \to \mathbb R)_{t\in \mathbbR}$ defined as $F_t(x)=F(x)+t$. Let ${\rm Trans}(F_t)$ be the translation number of $F_t$ defined by: \[{\rm Trans}(F_t) := \lim_{n\to +\infty}\frac{F_t^{\circ n}-{\rm Id}}{n}.\] Assume there is a Herman ring of modulus $2\tau$ associated to $F$ and let $p_n/q_n$ be the $n$-th convergent of ${\rm Trans}(F)$. Denoting $\ell_{\theta}$ as the length of the interval $\{t\in \mathbb R | {\rm Trans}(F_t)=\theta\}$, we prove that the sequence $(\ell_{p_n/q_n})$ decreases exponentially fast with respect to $q_n$. More precisely \[\limsup_{n \to \infty} \frac{1}{q_n} \log {\ell_{p_n/q_n}} \le -2\pi \tau .\]