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Parabolic fixed points, invariant curves and action-angle variables

Published online by Cambridge University Press:  19 September 2008

Dov Aharonov  and
Uri Elias
Dov Aharonov
Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel
Uri Elias
Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel
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Abstract

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A fixed point of an area-preserving mapping of the plane is called elliptic if the eigenvalues of its linearization are of unit modulus but not ±1; it is parabolic if both eigenvalues are 1 or −1. The elliptic case is well understood by Moser's theory. Here we study when is a parabolic fixed point surrounded by closed invariant curves. We approximate our mapping T by the phase flow of an Hamiltonian system. A pair of variables, closely related to the action-angle variables, is used to reduce T into a twist mapping. The conditions for T to have closed invariant curves are stated in terms of the Hamiltonian.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 2 , June 1990 , pp. 231 - 245
Copyright
Copyright © Cambridge University Press 1990

References

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