On the tip of the tongue
M. Misiurewicz and A. Rodrigues
Abstract
Recently we investigated the family of double standard maps of
the circle onto itself, given by
fa,b(x)=2x+a+(b/π)sin(2πx)
(mod 1). Similarly to the family of Arnold standard maps of the circle,
Aa,b(x)=x+a+(b/(2π))sin(2πx)
(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.