Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T00:53:36.975Z Has data issue: false hasContentIssue false
  • English
  • Français

A topological version of a theorem of Mather on twist maps

Published online by Cambridge University Press:  19 September 2008

Glen Richard Hall
Glen Richard Hall
Affiliation:
Department of Mathematics, Boston University, Boston, MA 02215, USA
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.

In this report we show that a twist map of an annulus with a periodic point of rotation number p/q must have a Birkhoff periodic point of rotation number p/q. We use topological techniques so no assumption of area-preservation or circle intersection property is needed. If the map is area-preserving then this theorem andthe fixed point theorem of Birkhoff imply a recent theorem of Aubry and Mather. We also show that periodic orbits of (significantly) smallest period for a twist map must be Birkhoff.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 4 , December 1984 , pp. 585 - 603
Copyright
Copyright © Cambridge University Press 1984

References

REFERENCES

[1]Asimov, D. & Franks, J.. Unremovable closed orbits. Preprint.Google Scholar
[2]Aubry, S.. Theory of the devil's staircase. Seminar on the Riemann Problem and Complete Integrability, 1978/1979, (Ed. Chudrovsky, D. O.), Lecture Notes in Math. 925 Springer: Berlin-Heidelberg-New York.Google Scholar
[3]Bernstein, D.. Birkhoff periodic orbits for twist maps with graph intersection property. Ergod. Th. & Dynam. Sys. To appear.Google Scholar
[4]Birkhoff, G. P.. Proof of Poincaré's geometric theorem. In George David Birkhoff: Collected Mathematical Papers, Vol. 1. Dover Publishing Inc: New York, 1968, p. 673.Google Scholar
[5]Birkhoff, G. D.. An extension of Poincaré's last geometric theorem. In George David Birkhoff: Collected Mathematical Papers, Vol. 2. Dover Publishing Inc: New York, 1968, p. 252.Google Scholar
[6]Block, L., Guckenheimer, J., Misureiwicz, M. & Young, L-S.. Periodic points and topological entropy of one dimensional maps. Preprint.Google Scholar
[7]Brown, M. & Neumann, W. D.. Proof of the Poincaré-Birkhoff fixed point theorem. Michigan Math. J. 24 (1977), 21–31.CrossRefGoogle Scholar
[8]Carter, P.. An improvement of the Poincaré-Birkhoff fixed point theorem. Trans. Amer. Math. Soc. 269 no. 1, (1982), 285–299.Google Scholar
[9]Chenciner, A.. Sur un enoncé dissipatif du théoremè geometric de Poincaré-Birkhoff. C. R. Acad. Sc. Paris, 294 Sec. 1, (1982), 243–245.Google Scholar
[10]Chenciner, A.. Bifurcations de points fixes elliptiques. Preprint.Google Scholar
[11]Hardy, G. H. & Wright, E. M.. An Introduction to the Theory of Numbers, (fifth edition). Oxford University Press: Oxford, 1979.Google Scholar
[12]Herman, M.. Sur la conjugaison differentiable des diffeomorphisms du cercle a des rotations. Publ. Math. I.H.E.S. 49 1979.CrossRefGoogle Scholar
[13]Herman, M.. Introduction a l'etude des courbes invariantes par les diffeomorphisms de l'anneau, Vol. 1. Preprint.Google Scholar
[14]Katok, A.. Some remarks on Birkhoff and Mather twist map theorems. Ergod. Th. & Dynam. Sys. 2 (1982), 185–192.CrossRefGoogle Scholar
[15]Mather, J. N.. Existence of quasi-periodic orbits for twist homomorphisms. Topology (1982).CrossRefGoogle Scholar
[16]Mather, J. N.. A criterion for the non-existence of invarian. circles. Preprint.Google Scholar
[17]Matsuoka, T.. The number and linking of periodic solutions of periodic systems. Invent. Math. 70 (1983), 319–340.CrossRefGoogle Scholar