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

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- by Kristian Bjerklöv and Tobias Jäger PDF
- J. Amer. Math. Soc.
**22**(2009), 353-362 Request permission

## 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.## References

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## Additional Information

**Kristian Bjerklöv**- Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G4
- Email: bjerklov@math.utoronto.ca
**Tobias Jäger**- Affiliation: Department of Mathematics, Collège de France, 3 rue d’Ulm, 75005 Paris, France
- Email: tobias.jager@college-de-france.fr
- Received by editor(s): August 10, 2006
- Published electronically: October 21, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**22**(2009), 353-362 - MSC (2000): Primary 37E45, 37B55
- DOI: https://doi.org/10.1090/S0894-0347-08-00627-9
- MathSciNet review: 2476777