On the widths of the Arnol’d tongues

Let $F: \mathbb{R} \rightarrow \mathbb{R} $ be a real analytic increasing diffeomorphism with $F- \mathrm{Id} $ being 1-periodic. Consider the translated family of maps $\mathop{({F}_{t} : \mathbb{R} \rightarrow \mathbb{R} )}\nolimits_{t\in \mathbb{R} } $ defined as ${F}_{t} (x)= F(x)+ t$. Let $\mathrm{Trans} ({F}_{t} )$ be the translation number of ${F}_{t} $ defined by $$\mathrm{Trans} ({F}_{t} ): = \lim _{n\rightarrow + \infty }\frac{{ F}_{t}^{\circ n} - \mathrm{Id} }{n} .$$ Assume that there is a Herman ring of modulus $2\tau $ associated to $F$ and let ${p}_{n} / {q}_{n} $ be the $n$th convergent of $\mathrm{Trans} (F)= \alpha \in \mathbb{R} \setminus \mathbb{Q} $. Denoting by ${\ell }_{\theta } $ the length of the interval $\{ t\in \mathbb{R} ~\mid ~\mathrm{Trans} ({F}_{t} )= \theta \} $, we prove that the sequence $({\ell }_{{p}_{n} / {q}_{n} } )$ decreases exponentially fast with respect to ${q}_{n} $. More precisely, $$\mathop {\mathrm{lim\hphantom{,}sup} }\limits _{n\rightarrow + \infty } \frac{1}{{q}_{n} } \log {\ell }_{{p}_{n} / {q}_{n} } \leq - 2\pi \tau .$$ There is a relation between ${\ell }_{{p}_{n} / {q}_{n} } $ and the width of the Arnol’d tongue, which confirms that the widths of the tongues decrease exponentially fast under suitable conditions.

CONSTRUCTING HERMAN RINGS BY TWISTING ANNULUS HOMEOMORPHISMS

Let F(z) be a rational map with degree at least three. Suppose that there exists an annulus $H \subset \widehat {\mathbb {C}}$ such that (1) H separates two critical points of F, and (2) F:H→F(H) is a homeomorphism. Our goal in this paper is to show how to construct a rational map G by twisting F on H such that G has the same degree as F and, moreover, G has a Herman ring with any given Diophantine type rotation number.

HERMAN RINGS OF BLASCHKE PRODUCTS OF DEGREE 3

Let Fa,λbe the Blaschke product of the form Fa,λ= λz2((z - a)/(1 - āz)) and α denote an irrational number satisfying the Brjuno condition. Henriksen [1997] showed that for any α there exists a constant a0≧ 3 and a continuous function λ(a) such that Fa,λ(a)possesses an Herman ring and also that modulus M(a) of the Herman ring approaches 0 as a approaches a0. It is remarked that the question whether a0= 3 holds or not is open. According to the idea of Fagella and Geyer [2003] we can show that for a certain set of irrational rotation numbers, a0is strictly larger than 3.