Genericity of mode-locking for quasiperiodically forced circle maps

TL;DR

This paper proves that for most quasiperiodically forced circle maps, the rotation number is rationally related and stable, leading to a devil's staircase structure in parameter space, highlighting prevalent mode-locking phenomena.

Contribution

It establishes the genericity of mode-locking in quasiperiodically forced circle homeomorphisms and describes the resulting structure of the rotation number function.

Findings

Most quasiperiodically forced circle maps are mode-locked.

The rotation number function forms a devil's staircase.

Mode-locking is stable under small perturbations.

Abstract

We show that a generic quasiperiodically forced circle homeomorphism is mode-locked: the rotation number in the fibres is rationally related to the rotation number in the base and it is stable under small perturbations of the system. As a consequence, this implies that for a generic parameter family of quasiperiodically forced circle homeomorphisms satisfying a twist condition, the graph of the rotation number as a function of the parameter is a devil's staircase.