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Birkhoff periodic orbits for twist maps with the graph intersection property

Published online by Cambridge University Press:  19 September 2008

David Bernstein
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720, USA
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Abstract

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In this paper we show that Birkhoff periodic orbits actually exist for arbitrary monotone twist maps satisfying the graph intersection property.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Aubry, S.. Theory of the Devil's staircase. In Seminar on the Riemann Problem and Complete Integrability, 19781979. (Ed. Chudnovsky, D. G.), Lecture Notes in Math. 925, Springer: Berlin-Heidelberg-New York.Google Scholar
[2]Birkhoff, G. D.. Proof of Poincare's geometric theorem. George David Birkhoff: Collected Mathematical Papers, vol. 1 Dover Publishing Inc: New York, 1968, p. 673.Google Scholar
[3]Carter, P.. An improvement of the Poincaré-Birkhoff fixed point theorem. Trans. Amer. Math. Soc. 269 no. 1, (1982), 285299.Google Scholar
[4]Hall, G. R.. A topological version of a theorem of Mather on twist maps. Ergod. Th. & Dynam. Sys. 4 (1984), 585603.CrossRefGoogle Scholar
[5]Katok, A.. Some remarks on Birkhoff and Mather twist map theorems. Ergod. Th. & Dynam. Sys. 2 (1982), 185194.CrossRefGoogle Scholar
[6]Katok, A.. Periodic and quasi-periodic orbits for twist maps. In Dynamical Systems and Chaos, Springer Lecture Notes in Physics 179 (1983) 4765.Google Scholar
[7]Katok, A.. More about Birkhoff periodic orbits and Mather sets for twist maps. Preprint.Google Scholar
[8]Mather, J. N.. Existence of quasi-periodic orbits for twist homeomorphisms. Topology (1982).CrossRefGoogle Scholar
[9]Moser, J.. Stable and Random Motions in Dynamical Systems. Princeton Univ. Press: Princeton, 1973.Google Scholar
[10]Rüssmann, H.. On the existence of invariant curves of twist mappings of an annulus. Preprint, 1982.CrossRefGoogle Scholar