The growth of the number of periodic orbits for annulus homeomorphisms and non-contractible closed geodesics on Riemannian or Finsler ℝP²

2022/11/20 by Hui Liu, Jian Wang, Liu, Hui +3
Mathematics · Physics and Astronomy · #37E30 #37E45 #Advanced Differential Geometry Research #Annulus (botany) #Astronomical and nuclear sciences #Combinatorics #Contractible space #Curvature #Dynamical Systems (math.DS) #FOS: Mathematics #Gaussian curvature #Geodesic #Geometric Analysis and Curvature Flows #Geometry #Mathematical analysis #Mathematics #Pure mathematics #Type (biology)

paper · pdf · doi:10.48550/arxiv.2211.10913

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2022/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we give a growth rate about the number of periodic orbits in the Franks type theorem obtained by the authors \citeLWY. As applications, we prove the following two results: there exist infinitely many distinct non-contractible closed geodesics on ℝP² endowed with a Riemannian metric such that its Gaussian curvature is positive, moreover, the number of non-contractible closed geodesics of length ≤ l grows at least like l²; and there exist either two or infinitely many distinct non-contractible closed geodesics on Finsler ℝP² with reversibility λ and flag curvature K satisfying (\fracλ1+λ)²

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