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Preprints Archive: Abstract of IC2010058 (2010)

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On the widths of the Arnol'd tongues

by Kuntal Banerjee

Document info: Pages 13, Figures 3.

Let $F: R \to R$ be a real analytic increasing diffeomorphism with $F-Id$ being 1 periodic. Consider the translated family of maps $(F_t :R \to R)_{t\in R}$ defined as $F_t(x)=F(x)+t$. Let ${\trans}(F_t)$ be the translation number of $F_t$ defined by [{\trans}(F_t) \:= \lim_{n\to +\infty}\frac{F_t^{\circ n}-{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 $\trans(F)=\alpha\in R\setminus \Q$. Denoting $\ell_{\theta}$ as the length of the interval ${t\in R~|~\trans(F_t)=\theta}$, 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].

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