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Conley–Zehnder index and bifurcation of fixed points of Hamiltonian maps

Published online by Cambridge University Press:  14 March 2017

YANXIA DENG  and
ZHIHONG XIA
YANXIA DENG
Affiliation:
Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada email [email protected]
ZHIHONG XIA
Affiliation:
Department of Mathematics, Southern University of Science and Technology, Shenzhen, PR China Department of Mathematics, Northwestern University, Evanston, IL 60208, USA email [email protected]
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Abstract

We study the bifurcations of fixed points of Hamiltonian maps and symplectic diffeomorphisms. We are particularly interested in the bifurcations where the Conley–Zehnder index of a fixed point changes. The main result is that when the Conley–Zehnder index of a fixed point increases (or decreases) by one or two, we observe that there are several bifurcation scenarios. Under some non-degeneracy conditions on the one-parameter family of maps, two, four or eight fixed points bifurcate from the original one. We give a relatively detailed analysis of the bifurcation in the two-dimensional case. We also show that higher-dimensional cases can be reduced to the two-dimensional case.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2086 - 2107
Copyright
© Cambridge University Press, 2017 

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