Abstract.
Recently we investigated the family of double standard maps of the circle onto itself, given by \(f_{a,b}(x) = 2x+a+\frac{b}{\pi}{\rm {sin}}(2\pi x)\) (mod 1). Similarly to the family of Arnold standard maps of the circle, \(A_{a,b}(x) = x+a+ \frac{b}{2\pi}{\rm {sin}}(2\pi x)\) (mod 1), if 0 < b ≤ 1 then any such map has at most one attracting periodic orbit. The values of the parameters for which such orbit exists are grouped into Arnold tongues. Here we study the shape of the boundaries of the tongues, especially close to their tips. It turns out that the shape is fairly regular, mainly due to the real analyticity of the maps.
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Dedicated to Vladimir Igorevich Arnold on the occasion of his 70th birthday
CMUP is supported by FCT through the POCTI and POSI programs, with Portuguese and European Community structural funds.
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Misiurewicz, M., Rodrigues, A. On the tip of the tongue. J. fixed point theory appl. 3, 131–141 (2008). https://doi.org/10.1007/s11784-008-0052-y
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DOI: https://doi.org/10.1007/s11784-008-0052-y