Geodesic
Rotation numbers for quasiperiodically forced circle maps-mode-locking vs. strict monotonicity
Bjerklöv, Kristian1 ; Jäger, Tobias2
1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G4
2 Department of Mathematics, Collège de France, 3 rue d’Ulm, 75005 Paris, France
Journal of the American Mathematical Society, Tome 22 (2009) no. 2, pp. 353-362

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-08-00627-9

Bjerklöv, Kristian 1 ; Jäger, Tobias 2

1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G4
2 Department of Mathematics, Collège de France, 3 rue d’Ulm, 75005 Paris, France 
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Bjerklöv, Kristian; Jäger, Tobias. Rotation numbers for quasiperiodically forced circle maps-mode-locking vs. strict monotonicity. Journal of the American Mathematical Society, Tome 22 (2009) no. 2, pp. 353-362. doi: 10.1090/S0894-0347-08-00627-9

[1] Herman, Michael-R. Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol′d et de Moser sur le tore de dimension 2 Comment. Math. Helv. 1983 453 502

[2] Jã¤Ger, Tobias H., Keller, Gerhard The Denjoy type of argument for quasiperiodically forced circle diffeomorphisms Ergodic Theory Dynam. Systems 2006 447 465

[3] Jã¤Ger, Tobias H., Stark, Jaroslav Towards a classification for quasiperiodically forced circle homeomorphisms J. London Math. Soc. (2) 2006 727 744

[4] Delyon, F., Souillard, B. The rotation number for finite difference operators and its properties Comm. Math. Phys. 1983 415 426

[5] Johnson, R., Moser, J. The rotation number for almost periodic potentials Comm. Math. Phys. 1982 403 438

[6] Prasad, Awadhesh, Negi, Surendra Singh, Ramaswamy, Ramakrishna Strange nonchaotic attractors Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2001 291 309

[7] Feudel, Ulrike, Kurths, Jã¼Rgen, Pikovsky, Arkady S. Strange non-chaotic attractor in a quasiperiodically forced circle map Phys. D 1995 176 186

[8] Glendinning, P., Feudel, U., Pikovsky, A. S., Stark, J. The structure of mode-locked regions in quasi-periodically forced circle maps Phys. D 2000 227 243

[9] Stark, J., Feudel, U., Glendinning, P. A., Pikovsky, A. Rotation numbers for quasi-periodically forced monotone circle maps Dyn. Syst. 2002 1 28

[10] Katok, Anatole, Hasselblatt, Boris Introduction to the modern theory of dynamical systems 1995

[11] Bjerklã¶V, Kristian Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations Ergodic Theory Dynam. Systems 2005 1015 1045

[12] Stark, J. Transitive sets for quasi-periodically forced monotone maps Dyn. Syst. 2003 351 364

[13] Furstenberg, H. Strict ergodicity and transformation of the torus Amer. J. Math. 1961 573 601

[14] Arnold, Ludwig Random dynamical systems 1998

[15] Johnson, Russell A. On a Floquet theory for almost-periodic, two-dimensional linear systems J. Differential Equations 1980 184 205

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