Abundance of Mode-Locking for Quasiperiodically Forced Circle Maps

Abstract

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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Correspondence to T. Jäger.

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Communicated by H.-T. Yau

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Wang, J., Jäger, T. Abundance of Mode-Locking for Quasiperiodically Forced Circle Maps. Commun. Math. Phys. 353, 1–36 (2017). https://doi.org/10.1007/s00220-017-2870-5

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