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Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces

Gonzalo Contreras, Marco Mazzucchelli

Abstract

We prove that a $C^2$-generic Riemannian metric on a closed surface has either an elliptic closed geodesic or an Anosov geodesic flow. As a consequence, we prove the $C^2$-stability conjecture for Riemannian geodesic flows of closed surfaces: a $C^2$-structurally stable Riemannian geodesic flow of a closed surface is Anosov. In order to prove these statements, we establish a general result that may be of independent interest and provides sufficient conditions for a Reeb flow of a closed 3-manifold to be Anosov.

Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces

Abstract

We prove that a -generic Riemannian metric on a closed surface has either an elliptic closed geodesic or an Anosov geodesic flow. As a consequence, we prove the -stability conjecture for Riemannian geodesic flows of closed surfaces: a -structurally stable Riemannian geodesic flow of a closed surface is Anosov. In order to prove these statements, we establish a general result that may be of independent interest and provides sufficient conditions for a Reeb flow of a closed 3-manifold to be Anosov.

Paper Structure

This paper contains 19 sections, 18 theorems, 126 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Anosov Reeb flows
  4. Organization of the paper
  5. Acknowledgements
  6. Surfaces of section in contact 3-manifolds
  7. Surfaces of section
  8. Non-degenerate closed Reeb orbits
  9. Open/broken book decompositions
  10. Size of images of the arrival map
  11. Hyperbolic invariant sets
  12. Hyperbolicity
  13. Local product structure
  14. A criterium for the Anosov property
  15. The closure of the space of periodic orbits
  16. ...and 4 more sections

Key Result

Theorem A

For any closed surface $M$, there exists a $C^2$-open dense subset $\mathcal{U}$ of the space of smooth Riemannian metrics on $M$ such that, for all $g\in\mathcal{U}$, one of the following two conditions is verified:

Figures (6)

  • Figure 1: (a) Foliation $\mathcal{F}|_D$ around a point $z$ on a radial binding component. (b) Foliation $\mathcal{F}|_D$ around a point $z$ on a broken binding component. In both pictures, the arrows are positively tangent to a projection of the Reeb vector field $X$ to $D$.
  • Figure 2: The path $\zeta_{r_0}$ accumulating on $S_0=\Sigma_{r_0}\cap W_\epsilon$.
  • Figure 3: Projections $z_{n_i}'$ to the unstable manifold $W^{u}(z')$.
  • Figure 4: The open set $U=U_1\cup W\cup U_2$ with an interval of heteroclinic intersections with boundary on $z_1$.
  • Figure 5: A heteroclinic rectangle.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Theorem A
  • Corollary 1.1
  • Theorem B
  • Theorem C
  • Theorem 1.2: Newhouse
  • Theorem D
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 28 more