Connectedness of the Arnold tongues for double standard maps

TL;DR

This paper proves that the Arnold tongues of double standard maps are connected, using complex quasiconformal techniques and analyzing the map's critical points.

Contribution

It introduces a novel proof of the connectedness of Arnold tongues for double standard maps via complex analysis methods.

Findings

Arnold tongues are connected for the family of double standard maps.

The proof employs quasiconformal techniques in the complex domain.

The complexification of the map has only one critical point, facilitating the proof.

Abstract

We show that the Arnold tongues of the family of double standard maps 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} has only one critical point taking symmetry into account.