The growth of the number of periodic orbits for annulus homeomorphisms and non-contractible closed geodesics on Riemannian or Finsler $\mathbb{R}P^2$

TL;DR

This paper investigates the growth rate of the number of non-contractible closed geodesics on real projective plane surfaces with Riemannian or Finsler metrics, establishing quadratic growth under certain curvature conditions.

Contribution

It extends previous results by providing new growth rate estimates for non-contractible closed geodesics on $ ext{RP}^2$ with specific geometric conditions.

Findings

Existence of infinitely many non-contractible closed geodesics on positively curved Riemannian $ ext{RP}^2$

Number of such geodesics of length ≤ l grows at least like l^2

Either two or infinitely many non-contractible closed geodesics exist on certain Finsler $ ext{RP}^2$ with quadratic growth rate

Abstract

In this article, we give a growth rate about the number of periodic orbits in the Franks type theorem obtained by the authors \cite{LWY}. As applications, we prove the following two results: there exist infinitely many distinct non-contractible closed geodesics on $\mathbb{R}P^2$ endowed with a Riemannian metric such that its Gaussian curvature is positive, moreover, the number of non-contractible closed geodesics of length $\leq l$ grows at least like $l^2$; and there exist either two or infinitely many distinct non-contractible closed geodesics on Finsler $\mathbb{R}P^2$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$, furthermore, if the second case happens, then the number of non-contractible closed geodesics of length $\leq l$ grows at least like $l^2$.