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Quadratic growth of geodesics on the two-sphere

Bernhard Albach

Abstract

We prove that for any reversible Finsler metric on S2, the number of prime closed geodesics grows quadratically with respect to length. The main tools are an improvement on Franks' theorem about the number of periodic points of area-preserving annulus maps, and the theory of cylindrical contact homology in the complement of a link.

Quadratic growth of geodesics on the two-sphere

Abstract

We prove that for any reversible Finsler metric on S2, the number of prime closed geodesics grows quadratically with respect to length. The main tools are an improvement on Franks' theorem about the number of periodic points of area-preserving annulus maps, and the theory of cylindrical contact homology in the complement of a link.

Paper Structure

This paper contains 23 sections, 36 theorems, 137 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Dynamics of annulus maps
  3. Cylindrical contact homology
  4. Conley-Zehnder index
  5. Chain complex
  6. Differential
  7. Chain map
  8. An explicit chain map
  9. Splitting
  10. The model system
  11. Sphere of revolution
  12. The model system
  13. Unit tangent bundle
  14. Hilbert contact form
  15. Clairaut integral
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $g$ be a reversible Finsler metric on $S^2$. Then the growth rate of $P^t(g)$ is at least quadratic:

Figures (4)

  • Figure 1: The model sphere of rotation
  • Figure 2: The angle $\beta$
  • Figure 3: Euler angles
  • Figure 4: The link $L_1$ projected to $\mathbb{R}^3$

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Franks MR0951509,MR1161099,MR0967632
  • Lemma 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Conjecture 1.7: Hryniewicz
  • Theorem 2.1
  • Theorem 2.2: Franks MR1161099
  • Lemma 2.3
  • ...and 66 more