Quasigeodesic Anosov flows and homotopic properties of flow lines
The author uses tools from topology and dynamical systems to give a detailed analysis of the global structure of a quasigeodesic Anosov flow \(\varphi\) on a compact 3-manifold \(M\) whose fundamental group \(\pi_1 (M)\) is Gromov-hyperbolic. Here a flow is called quasi-geodesic if the lift to the universal covering \(\widetilde {M}\) of \(M\) of any orbit of \(\varphi\) in \(M\) is a quasi-geodesic with respect to any \(\pi_1 (M)\)- periodic metric. It is shown that such flows are fundamentally different from suspensions of Anosov-diffeomorphisms or geodesic flows on surfaces. Namely the leaf-space \({\mathcal H}^s\) of the stable foliation \({\mathcal F}^s\) in \(\widetilde {M}\) is not Hausdorff. The number of non- Hausdorff points in \({\mathcal H}\) is finite, and every such point corresponds to a non-trivial free homotopy between two different closed orbits of \(\varphi\) in \(M\). The limit set \(\Lambda_F\) of any leaf \(F\) of \({\mathcal F}^s\) in the ideal boundary \(S^2_\infty\) of \(\widetilde {M}\) is neither a Jordan curve, nor the whole sphere. Again, an identification in \(\Lambda_F\) of two distinct points in the intrinsic boundary of \(F\) corresponds to an end-point in \(S^2_\infty\) of lift of two freely homotopic closed orbits of \(\varphi\) in \(M\).
non-trivial free homotopy between closed orbits
scientific article; zbMATH DE number 761171