On the widths of the Arnol'd Tongues

TL;DR

This paper investigates the widths of Arnold tongues for a class of real analytic circle diffeomorphisms, proving that these widths decrease exponentially fast for certain rational rotation numbers related to Herman rings.

Contribution

It establishes an exponential decay rate for the widths of Arnold tongues associated with Herman rings in analytic circle diffeomorphisms.

Findings

Widths of Arnold tongues decrease exponentially with respect to denominator q_n.

The decay rate is bounded above by -2π times the Herman ring modulus.

The result links geometric properties of Herman rings to dynamical features of circle maps.

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 \[\limsup_{n \to \infty} \frac{1}{q_n} \log {\ell_{p_n/q_n}} \le -2\pi \tau .\]