Quasigeodesic Anosov flows and homotopic properties of flow lines
type: Item
instance of: scholarly article
title: Quasigeodesic Anosov flows and homotopic properties of flow lines
zbMATH Open document ID: 0832.57009
author: Sergio R. Fenley
published in: Journal of Differential Geometry
publication date: 1996-03-07T00:00:00Z
review text: 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 \(M\).
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 37D99
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 761171
zbMATH Keywords: hyperbolic group
zbMATH Keywords: non-trivial free homotopy between closed orbits
zbMATH Keywords: leaf- space of the stable foliation
zbMATH Keywords: quasigeodesic Anosov flow
zbMATH Keywords: compact 3- manifold
zbMATH Keywords: fundamental group
zbMATH Keywords: non-Hausdorff points
zbMATH Keywords: limit set
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214456224
OpenAlex ID: W1562404870
Wikidata QID: Q115178575
DOI: 10.4310/JDG/1214456224
description: scientific article; zbMATH DE number 761171