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Regular dependence of invariant curves and Aubry–Mather sets of twist maps of an annulus

Published online by Cambridge University Press:  19 September 2008

Raphaël Douady
Raphaël Douady
Affiliation:
Ecole Polytechnique, Centre de Mathématiques, 91128 Palaiseau Cedex, France
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Abstract

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We prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets.

Then we show that invariant curves of the same map with different rotation numbers ω and ω′ cannot approach each other at a distance less than cst. |ω−ω′|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure.

Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Pöschel (for the second one), in the case of Hamiltonian systems and area preserving maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 4 , December 1988 , pp. 555 - 584
Copyright
Copyright © Cambridge University Press 1988

References

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