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

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 $l^2$.