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Minimization and Hyperbolicity

Gonzalo Contreras, Daniel Offin

Abstract

In this paper we study the relationship between the strict locally minimizing orbits for time dependent lagrangian systems and hyperbolicity properties of the corresponding lagrangian flow.

Minimization and Hyperbolicity

Abstract

In this paper we study the relationship between the strict locally minimizing orbits for time dependent lagrangian systems and hyperbolicity properties of the corresponding lagrangian flow.

Paper Structure

This paper contains 11 sections, 6 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Autonomous Lagrangians.
  3. The time-dependent case.
  4. The hamiltonian Jacobi and Riccati equations.
  5. Conjugate points and Green bundles.
  6. The index form
  7. The autonomous case.
  8. The transversal behaviour.
  9. The graph transform.
  10. Hyperbolicity.
  11. Quasi Hyperbolicity

Key Result

Theorem A

Let $k\in{\mathbb R}$ and $\Lambda\subset E^{-1}\{k\}$ be a compact $\varphi_t$-invariant set. If there is $a>0$ such that for all $(\theta,T)\in\Lambda\times{\mathbb R}^+$ and $\xi\in\Omega(\theta,T)$: then $\Lambda$ is hyperbolic.

Theorems & Definitions (17)

  • Theorem A
  • Corollary B
  • Theorem C
  • Theorem A1
  • Corollary B1
  • Theorem C1
  • proof
  • proof
  • proof
  • proof
  • ...and 7 more