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

Citation

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. https://resolver.caltech.edu/CaltechTHESIS:11122012-095207855

Abstract

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.

Thesis Files