Quadratic growth of geodesics on the two-sphere

TL;DR

This paper proves that on the two-sphere with any reversible Finsler metric, the count of prime closed geodesics increases quadratically with their length, using advanced topological and dynamical tools.

Contribution

It establishes a quadratic growth rate for prime closed geodesics on S2 under reversible Finsler metrics, extending previous understanding with new topological methods.

Findings

Prime closed geodesics grow quadratically with length

Improved bounds on periodic points of area-preserving maps

Application of cylindrical contact homology techniques

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.