On the tip of the tongue

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.