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On the widths of the Arnol’d tongues

Published online by Cambridge University Press:  03 May 2013

KUNTAL BANERJEE*
Affiliation:
Mathematics Section, The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy email [email protected] Current address: LAMA, Université Paris-Est Créteil, 61 avenue du Général de Gaulle, 94010 Créteil, France.

Abstract

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.

Type
Research Article
Copyright
Copyright ©2013 Cambridge University Press 

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References

Arnol’d, V. I.. Small denominators. I. Mappings of the circumference onto itself. Amer. Math. Soc. Transl. Ser. 2 46 (1965), 213284.Google Scholar
Banerjee, K.. On the Arnol’d tongues for circle homeomorphisms. PhD Thesis, Université Paul Sabatier – Toulouse III, 2010.Google Scholar
Denjoy, A.. Sur les courbes définies par les équations différentielles à la surface du tore. J. Math. Pures Appl. (9) 11 (1932), 333375.Google Scholar
Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, New York, 1980.Google Scholar
Herman, M. R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5233.Google Scholar
Herman, M. R.. Mesure de Lebesgue et nombre de rotation. Geometry and Topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) (Lecture Notes in Mathematics, 597). Springer, Berlin, 1977, pp. 271293.Google Scholar
Poincaré, H.. Mémoire sur les courbes définies par une équation différrentielle III. J. Math. Pures Appl. 4 (1885), 167244.Google Scholar
Yoccoz, J.-C.. Analytic linearization of circle diffeomorphisms. Dynamical Systems and Small Divisors (Cetraro, 1998) (Lecture Notes in Mathematics, 1784). Springer, Berlin, 2002, pp. 125173.Google Scholar