Communications in Mathematical Physics

, Volume 353, Issue 1, pp 1–36 | Cite as

Abundance of Mode-Locking for Quasiperiodically Forced Circle Maps

  • J. Wang
  • T. JägerEmail author


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 \({\mathcal{C}^1}\)-open condition on the geometry of twist parameter families of such systems, the closure of the union of mode-locking 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 \({{\rm SL}(2,\mathbb{R})}\)-cocycles by Young and Bjerklöv. The methods apply to a variety of examples, including a forced version of the classical Arnold circle map.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Science and TechnologyNanjingChina
  2. 2.Institute of Mathematics, Friedrich-Schiller-University JenaJenaGermany

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