Connectedness of the Arnold tongues for double standard maps

Abstract

We show that Arnold tongues for the family of double standard maps \[ f a , b ( x ) = 2 x + a − ( b / π ) s i n ( 2 π x ) f_{a,b}(x)=2x+a-(b/\pi )sin(2 \pi x) \] are connected. This proof is accomplished in the complex domain by means of quasiconformal techniques and depends partly upon the fact that the complexification of f a , b f_{a,b} , has only one critical orbit taking symmetry into account.