On the dynamics of uniform Finsler manifolds of negative flag curvature
Let \(\widetilde M\) denote a complete, simply connected Riemannian manifold with sectional curvature bounded above and below by negative constants, and let \(\widetilde M (\infty)\) denote the boundary at infinity consisting of equivalence classes of asymptotic unit speed geodesics of \(\widetilde M\). There is a natural topology on \(\widetilde M(\infty)\) that makes it homeomorphic with \(S_p\widetilde M\), the unit vectors of \(\widetilde M\) at \(p\), where \(p\) is any point of \(\widetilde M\). In the course of solving the Dirichlet problem for harmonic functions on \(\widetilde M\) with prescribed boundary values on \(\widetilde M(\infty)\), \textit{M. T. Anderson} and \textit{R. Schoen} [Ann. Math., II. Ser. 121, 429-461 (1985; Zbl 0587.53045)] proved that \(\widetilde M(\infty)\) admits a Hölder structure. In this paper the author proves the existence of a Hölder structure on \(\widetilde M(\infty)\) in the case that \(\widetilde M\) is a reversible Finsler manifold with Anosov geodesic flow and negative flag curvature relative to a natural connection on \(H\widetilde M= (T\widetilde M- \{0\})/ \mathbb{R}^+\), provided that the quotient space \(M/I(\widetilde M)\) with the quotient topology is compact. Here \(I(\widetilde M)\) denotes the isometry group of \(\widetilde M\). The space \(\widetilde M(\infty)\) is identified with the set of stable manifolds \(\{{\mathcal F}^S(v),\;v\in H\widetilde M\}\), and geometric properties known for Riemannian manifolds of strictly negative sectional curvature are generalized to this Finsler setting. Using some of the arguments of this paper, the author also observes that the statements and proofs of certain results due to M. Gromov, E. Dinaburg and A. Manning remain valid in the Finsler case. In particular, (1) (Gromov) If \(M\) and \(N\) are compact Finsler manifolds of negative flag curvature with isomorphic fundamental groups, then there is a homeomorphism between \(HM\) and \(HN\) that preserves the geodesic foliations; (2) (Dinaburg, Manning) Let \(M\) be a compact Finsler manifold. Then the volume entropy of \(M\) is at most the topological entropy of \(M\), and equality holds if \(M\) has nonpositive flag curvature.
Hölder structure on the boundary at infinity
Finsler manifolds of negative flag curvature
scientific article; zbMATH DE number 1014195