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

Authors: Kristian Bjerklöv and Tobias Jäger
Journal: J. Amer. Math. Soc. 22 (2009), 353-362
MSC (2000): Primary 37E45, 37B55
Published electronically: October 21, 2008
MathSciNet review: 2476777
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Abstract: We describe the relation between the dynamical properties of a quasiperiodically forced orientation-preserving circle homeomorphism $f$ 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 $f$ 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.

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Additional Information

Kristian Bjerklöv
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G4

Tobias Jäger
Affiliation: Department of Mathematics, Collège de France, 3 rue d’Ulm, 75005 Paris, France

Keywords: Rotation numbers, mode-locking, quasiperiodically forced systems.
Received by editor(s): August 10, 2006
Published electronically: October 21, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.