Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres

TL;DR

This paper establishes conditions under which a bumpy Finsler 3-sphere must have multiple closed geodesics, either two non-hyperbolic or at least three, contributing to the understanding of geodesic multiplicity and stability.

Contribution

It proves a new multiplicity result for closed geodesics on bumpy Finsler 3-spheres, highlighting the existence of multiple geodesics under specific conditions.

Findings

Either two non-hyperbolic prime closed geodesics exist

Or there are at least three prime closed geodesics

Results apply to Q-homological Finsler 3-spheres with bumpy, irreversible metrics

Abstract

We prove that for every $\Q$-homological Finsler 3-sphere $(M,F)$ with a bumpy and irreversible metric $F$, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics.