On some problems in Lagrangian Dynamics and Finsler Geometry

Open Access
Author:
Chen, Dong
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
March 15, 2017
Committee Members:
  • Dmitri Yu Burago, Dissertation Advisor
  • Dmitri Yu Burago, Committee Chair
  • Federico Juan Rodriguez Hertz, Committee Member
  • Mark Levi, Committee Member
  • Runze Li, Outside Member
Keywords:
  • KAM theory
  • Finsler metric
  • duel lens map Hamiltonian flow
  • perturbation
  • metric entropy
  • duel lens map
  • Hamiltonian flow
Abstract:
The purpose of this dissertation is to present several applications of enveloping functions and dual lens maps to geometry and dynamical systems. In Chapter 1 we have a brief review on basic notions and theory we need to understand the main results. In Chapter 2 we prove that given a point on a Finsler surface, one can always find a neighborhood of the point and isometrically embed this neighborhood into a Finsler torus without conjugate points. The major tool is enveloping functions. In Chapter 3 we introduce the dual lens map technique developed by Burago and Ivanov. It derives from enveloping functions and symplectic geometry. We then show how this technique is used to perturb the geodesic flows of flat Finsler tori. In Chapter 4 we show how dual lens map can be used in KAM theory. The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. We show two types of Lagrangian perturbations of the geodesic flow on flat Finsler tori. The perturbations are $C^\infty$ small but the resulting flows has a positive measure of trajectories with positive Lyapunov exponent. The measure of this set is of course extremely small. Still, the flow has positive metric entropy. From this result we get positive metric entropy outside some KAM tori and it gives positive answer to a question asked by Kolmogorov.