Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity
type: Item
instance of: scholarly article
title: Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity
author: Gerhard Knieper
author: Q6063420
published in: Proceedings of the American Mathematical Society
publication date: 2024-09-11T00:00:00Z
review text: Let \(M\) be a compact manifold, without boundary, and let \({\mathscr F}_{\mathrm{hyp}} (M)\) denote that set of Finsler metrics on \(M\) all of whose closed geodesics are hyperbolic. The paper under review is mainly devoted to showing that the interior of \({\mathscr F}_{\mathrm{hyp}} (M)\) (with respect to the \(C^2\) topology) consists of the Finsler metrics whose geodesic flow is Anosov. An ingredient in the proof is a characterization of Anosov metrics among the Finslerian metrics without conjugate points, first established within the Riemannian category by \textit{R. O. Ruggiero} [Math. Z. 208, No. 1, 41--55 (1991; Zbl 0749.58042)] and successively extended to the Finslerian category by \textit{G. Contreras} et al. [Nonlinearity 11, No. 2, 355--361 (1998; Zbl 0999.37037)]. \N\NReviewer's remark. The authors' research may profit, at least from a cultural perspective, from a comparison to the more classical contributions to Finslerian geometry, see, e.g., [\textit{P. Dazord}, Bull. Soc. Math. Fr. 99, 171--192 (1971; Zbl 0215.23504)].
reviewed by: Sorin Dragomir
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37A10
Mathematics Subject Classification ID: 53B40
Mathematics Subject Classification ID: 53C60
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 7910781
zbMATH Keywords: Finsler metric
zbMATH Keywords: conjugate point
zbMATH Keywords: Anosov flow
MaRDI profile type: Publication
cites work: Q3128806
cites work: Lagrangian graphs, minimizing measures and Mañé's critical values
cites work: On the creation of conjugate points for Hamiltonian systems
cites work: Jets of closed orbits of Mañé's generic Hamiltonian flows
cites work: Proof of the \(C^2\)-stability conjecture for geodesic flows of closed surfaces
cites work: Q5705858
cites work: When is a geodesic flow of Anosov type, II
cites work: Riemannian manifolds with geodesic flow of Anosov type
cites work: Q4181002
cites work: Q4474934
cites work: Generic properties of geodesic flows
cites work: Q4130683
cites work: Quasi-Elliptic Periodic Points in Conservative Dynamical Systems
cites work: On Anosov energy levels of convex Hamiltonian systems
cites work: The second closed geodesic, the fundamental group, and generic Finsler metrics
cites work: On the creation of conjugate points
cites work: Q4070368
cites work: Hamiltonian systems: Generic properties of closed orbits and local perturbations
DOI: 10.1090/PROC/16423
description: scientific article; zbMATH DE number 7910781