Genericity of mode-locking for quasiperiodically forced circle maps

Abundance of mode-locking for quasiperiodically forced circle maps

We study the phenomenon of mode-locking in the context of quasiperiodically forced non-linear circle maps. As a main result, we show that under certain C1-open condition on the geometry of twist parameter families of such systems, the closure of the union of modelocking plateaus has positive measure. In particular, this implies the existence of infinitely many mode-locking plateaus (open Arnold tongues). The proof builds on multiscale analysis and parameter exclusion methods in the spirit of Benedicks and Carleson, which were previously developed for quasiperiodic SL(2,R)-cocycles by Young and…

拟周期强迫圆映射的旋转数--锁模与严格单调性

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

本文描述了在严格单调扰动下，拟周期强迫保向圆同胚的动力学性质与纤维旋转数的行为之间的关系。尽管在强迫情况下的动力学可能要复杂得多，但我们得到的结果与一维情况完全相似。特别是，纤维旋转数的行为严格单调时，旋转向量是无理的。这回答了Herman（1983，Comm. Math. Helv.）提出的一个问题。

Linearization of quasiperiodically forced circle flow beyond Brjuno condition

We prove that an analytic quasiperiodically forced circle flow with a not super-Liouvillean base frequency and which is close enough to some constant rotation is $C^{\infty}$ rotations reducible, provided its fibered rotation number is Diophantine with respect to the base frequency. As a corollary, we obtain that among such systems, the linearizable ones and those displaying mode-locking are locally dense for the $C^\infty$-topology.

On topological genericity of the mode-locking phenomenon

We study the circle homeomorphisms extensions over a strictly ergodic homeomorphism. Under a very mild restriction, we show that the fibered rotation number is locally constant on an open and dense subset. In the complement of this set, we found a dense subset in which every map is conjugate to a direct product. Our result provides a generalisation of Avila-Bochi-Damanik's result on ${\rm SL}(2,\mathbb{R})-$cocycles, and Jager-Wang-Zhou's result on quasi-periodically forced maps, to a broader setting.