Rotation numbers for quasiperiodically forced circle maps-mode-locking vs. strict monotonicity

Abstract

We describe the relation between the dynamical properties of a quasiperiodically forced orientation-preserving circle homeomorphism f f and the behaviour of the fibred rotation number with respect to strictly monotone perturbations. Despite the fact that the dynamics in the forced case can be considerably more complicated, the result we obtain is in perfect analogy with the one-dimensional situation. In particular, the fibred rotation number behaves strictly monotonically whenever the rotation vector of f f is irrational, which answers a question posed by Herman (1983). In addition, we obtain the continuous structure of the Arnold tongues in parameter families such as the quasiperiodically forced Arnold circle map.