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Birkhoff periodic orbits, Aubry-Mather sets, minimal geodesics and Lyapunov exponents


Chen, Wei-Feng (1993) Birkhoff periodic orbits, Aubry-Mather sets, minimal geodesics and Lyapunov exponents. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/g2ca-jx87.


Aubry-Mather theory proved the existence of invariant circles and invariant Cantor set (the ghost circles) for the area-preserving, monotone twist maps of annulus or of cylinders. We are interested in higher dimensional systems. The celebrated KAM theorem established the existence of invariant tori for small perturbations of integrable Hamiltonian systems with nondegenerate Hamiltonian functions, but said nothing about the missing tori. Bernstein-Katok found the Birkhoff periodic orbits, which are viewed as the traces of missing tori, for the system in the KAM theorem but under the stronger condition that the Hamiltonian function is convex. We find the "isolating block", a structure invented by Conley and Zehnder, to demonstrate the existence of Birkhoff periodic orbits for the KAM system.

In the second part, we wanted to establish the existence of minimal closed geodesic which is hyperbolic on the surface of genus greater than one. There is strong evidence that such geodesics exist. We find a curvature condition for the minimal closed geodesic, thus furnishing further evidence.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Katok, Anatole
Thesis Committee:
  • Unknown, Unknown
Defense Date:10 June 1992
Record Number:CaltechTHESIS:11122012-095207855
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7264
Deposited By: Benjamin Perez
Deposited On:13 Nov 2012 16:35
Last Modified:09 Nov 2022 19:20

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