Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity (Q6602157)

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)].