Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rigidity of Lyapunov Exponents for Geodesic Flows

Nestor Nina Zarate, Sergio Romaña

Abstract

In this paper, we study rigidity problems between Lyapunov exponents along periodic orbits and geometric structures. More specifically, we prove that for a surface M without focal points, if the value of the Lyapunov exponents is constant over all periodic orbits, then $M$ is the flat 2-torus or a surface of constant negative curvature. We obtain the same result for the case of Anosov geodesic flow for surface, which generalizes Butler's result in dimension two. Using completely different techniques, we also prove an extension of Butler's result to the finite volume case, where the value of the Lyapunov exponents along all periodic orbits is constant, being the maximum or minimum possible.

Rigidity of Lyapunov Exponents for Geodesic Flows

Abstract

In this paper, we study rigidity problems between Lyapunov exponents along periodic orbits and geometric structures. More specifically, we prove that for a surface M without focal points, if the value of the Lyapunov exponents is constant over all periodic orbits, then is the flat 2-torus or a surface of constant negative curvature. We obtain the same result for the case of Anosov geodesic flow for surface, which generalizes Butler's result in dimension two. Using completely different techniques, we also prove an extension of Butler's result to the finite volume case, where the value of the Lyapunov exponents along all periodic orbits is constant, being the maximum or minimum possible.
Paper Structure (11 sections, 14 theorems, 54 equations)

This paper contains 11 sections, 14 theorems, 54 equations.

Table of Contents

  1. Introduction.
  2. Preliminaries and Notation
  3. Geodesic Flow
  4. No conjugate points and No focal Points
  5. Green Subbundles
  6. Jacobi Tensor and Riccati Equation
  7. Lyapunov Exponents
  8. Rigidity in dimension 2.
  9. Rigidity Theorems
  10. Rigidity of Lyapunov exponents in finite volume case.
  11. Proof of the Theorem \ref{['thm4.5']}

Key Result

Theorem 1.1

Bu Let $M$ be an $m$-dimensional compact negatively curved Riemannian manifold. Suppose that for every periodic point $\theta$ of the geodesic flow $\phi:SM\to SM$. Then $M$ is homothetic to a compact quotient of $\mathbb{H}_{\mathbb{R}}^{m}.$

Theorems & Definitions (28)

  • Theorem 1.1
  • Conjecture 1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 3.1
  • Definition 3.2
  • Theorem 3.3
  • ...and 18 more