On the widths of the Arnol'd Tongues
Kuntal Banerjee

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 be a real analytic increasing diffeomorphism with being 1 periodic. Consider the translated family of maps defined as . Let be the translation number of 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 associated to and let be the -th convergent of . Denoting as the length of the interval , we prove that the sequence decreases exponentially fast with respect to . More precisely \[\limsup_{n \to \infty} \frac{1}{q_n} \log {\ell_{p_n/q_n}} \le -2\pi \tau .\]
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Taxonomy
TopicsMathematical Dynamics and Fractals · Molecular spectroscopy and chirality · Liquid Crystal Research Advancements
