Hostname: page-component-89b8bd64d-9prln Total loading time: 0 Render date: 2026-05-08T01:31:38.467Z Has data issue: false hasContentIssue false
  • English
  • Français

Propriétés des attracteurs de Birkhoff

Published online by Cambridge University Press:  19 September 2008

P. Le Calvez
P. Le Calvez
Affiliation:
Université Paris-Sud et UA1169 du CNRS Mathématiques, Batiment 425, F-91405 Orsay 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 study dissipative twist maps of the annulus, following the ideas of G. D. Birkhoff explained in an article of 1932.

In the first part, we give complete and rigorous proofs of the results of this article. We define the Birkhoff attractor of a dissipative twist map which has an attracting bounded annulus, we give its main properties and we define its upper and lower rotation numbers.

In the second part we give further results on these sets, thus we show that they often coincide with the closure of a hyperbolic periodic point and that they can contain an infinite number of sinks. We also show that the Birkhoff attractors don't depend on a continuous way on the maps.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 2 , June 1988 , pp. 241 - 310
Copyright
Copyright © Cambridge University Press 1988

References

BIBLIOGRAPHIE

[AK]Auslander, J. & Katznelson, Y.. Continuous maps of the circle without periodic points. Israël J. Math. 32 (1979), 375–381.CrossRefGoogle Scholar
[BG]Berger, M. & Gastiaux, B.. Géométrie Différentielle. Armand Colin (1972), Paris.Google Scholar
[B1]Birkhoff, G. D.. Sur quelques courbes fermées remarquables, Bull. Soc. Math. France 60 (1932), 1–26;Google Scholar
aussi dans Collected Math. Papers of G. D. Birkhoff, vol. II, p. 418–443, Dover, New York (1968).Google Scholar
[B2]Birkhoff, G. D.. Sur l'existence de régions d'instabilité en dynamique, Ann. Inst. Henri Poincaré, 8 (1932) et Collected Math. Papers, vol. II, p. 444–461.Google Scholar
[Cha1]Charpentier, M.. Sur quelques propriétés des courbes de M. Birkhoff. Bull. Soc. Math. France 62 (1934), 193–224.Google Scholar
[Cha2]Charpentier, M.. Sur des courbes fermées analogues aux courbes de M. Birkhoff. J. Math. Pures et Appl. 14 (1935), 1–48.Google Scholar
[Che]Chenciner, A.. Séminaire Bourbaki, no 622, Astérisque 121–123Soc. Math. de France (1985), 147–170.Google Scholar
[CGT]Chenciner, A., Gambaudo, J. M. & Tresser, C.. Une remarque sur la structure des endomorphismes de degré 1 du cercle. C.R. Acad. Sc. Paris, 299, série I, no 5 (1984), 145–147.Google Scholar
[Cho]Choquet, G.. Lectures on Analysis, vol. 1, W. A. Benjamin, Inc (1969), New York.Google Scholar
[G]Goroff, D. L.. Hyperbolic sets for twist maps. Ergod. Th. & Dynam. Sys. 5 (1985), 337–354.CrossRefGoogle Scholar
[Ha]Hall, G. R.. A topological version of a theorem of Mather on twist maps. Ergod. Th. & Dynam. Sys. 4 (1984), 585–603.CrossRefGoogle Scholar
[He1]Herman, M.. Sur la conjugaison différentiable des difféomorphismes du cercle à une rotation. Publ. Math. IHES, 49 (1979), 5–234.CrossRefGoogle Scholar
[He2]Herman, M.. Sur les courbes invariantes par les difféomorphismes de l'anneau. Asterisque 103–104, Soc. Math. France (1983).Google Scholar
[HPS]Hirsch, M. W., Pugh, C. C. & Shub, M.. Invariant Manifolds. Lecture Notes Math., vol. 583, Springer, New York.Google Scholar
[Ka]Katok, A.. Some remarks on Birkhoff and Mather twist map theorem. Ergod. Th. & Dynam. Sys. 2 (1982), 185–194.CrossRefGoogle Scholar
[Ku]Kuratowski, C.. Topologie, Vol. I et II. Warszawa (1961).Google Scholar
[L]Le Calvez, P.. Existence d'orbites quasi-périodiques dans les attracteurs de Birkhoff. Commun. Math. Phys. 106 (1986), 383–394.CrossRefGoogle Scholar
[Ma]Markushevitch, A. I.. Theory of Functions of a Complex Variable, vol. III. Prentice Hall (1967).Google Scholar
[Mi]Misiurewicz, M.. Twist sets for maps of the circle. Ergod. Th. & Dynam. Sys. 4 (1984), 391–404.CrossRefGoogle Scholar
[Ne]Nehari, Z.. Conformal Mappings. McGraw-Hill (1975), New York.Google Scholar
[Newh]Newhouse, S. E.. The abundance of wild hyperbolic sets and non smooth stable sets for diffeomorphisms. Publ. Math. IHES, 50 (1979), 101–152.CrossRefGoogle Scholar
[NPT]Newhouse, S., Palis, J. & Takens, F.. Bifurcations and stability of families of diffeomorphisms. Publ. Math. IHES, 57 (1983), 5–72.CrossRefGoogle Scholar
[Newm]Newman, M. H. A.. Elements of the Topology of Plane Sets of Points. Cambridge University Press (1939), Cambridge.Google Scholar
[Ni]Nitecki, Z.. Differentiable Dynamics. The MIT Press (1971), Cambridge USA.Google Scholar