Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-07-06T14:17:51.546Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

[1]Aubry, S. & Le Daeron, P. Y.. The discrete Frenkel-Kontorova model and its extensions I. Physica 8D (1983), 381–422.Google Scholar
[2]Bost, J-B.. Tores invariants des systèmes dynamiques hamiltoniens. Séminaire Bourbaki 639 (02. 1985); Astérisque 133–134 (1986).Google Scholar
[3]Chenciner, A.. La dynamique au voisinage d'un point fixe elliptique de Poincaré et Birkhoff à Aubry et Mather. Séminaire Bourbaki 622 (02. 1984), published in Astérisque 121–122 (1985) 147–170.Google Scholar
[4]Hedlund, G.. Geodesies on a 2-dimensional riemannian manifold with periodic coefficients. Ann. Math. serie II 33 (1932).Google Scholar
[5]Herman, M. R.. Sur la conjugation différentiable des difféomorphismes du cercle à des rotations. I.H.E.S. Pub. Math. 49 (1975), 5–234.Google Scholar
[6]Herman, M. R.. Démonstration du théorème de la courbe translatée. Manuscript, 1980.Google Scholar
[7]Herman, M. R.. Sur les courbes invariantes par les difféomorphismes de l'anneau. 1Astérisque 103–104 (1983) Astérisque 144 (1986).Google Scholar
[8]Herman, M. R.. Remarque sur les courbes de Mather et Aubry. Conference at the séminaire de l'Ecole Polytechnique,14/2/1983.Google Scholar
[9]Lazutkin, V. F.. The existence of caustics for a billiard problem in a convexe domain. Math. USSR-Iszvestija 71 (1973), 185–214.CrossRefGoogle Scholar
[10]Mather, J.. Existence of quasi-periodic orbits for twist homeomorphisms. Topology 21, 4 (1982), 457–467.CrossRefGoogle Scholar
[11]Moser, J.. On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen (1962), 1–20.Google Scholar
[12]Pöschel, J.. Integrability of Hamiltonian systems on Cantor sets. Comm. Pure Appl. Math. 35 (1982), 653–696.Google Scholar
[13]Rüssmann, H.. On the existence of invariant curves of twist mappings of an annulus. Lect. Notes in Math. 1007 (1983), 677–718.Google Scholar
[14]Yoccoz, J-C.. Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Sci. E.N.S., 4e s. 17 (1984), 333–359.Google Scholar
[15]Zehnder, E.. Generalized implicit function theorem with application to some small divisors problems II. Comm. in Pure Appl. Math. 29 (1976), 49–111.Google Scholar