# Commentarii Mathematici Helvetici

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**Volume 73, Issue 2, 1998, pp. 259–297**

**DOI: 10.1007/s000140050055**

Published online: 1998-06-30

The structure of branching in Anosov flows of 3-manifolds

S. R. FenleyIn this article we study the topology of Anosov flows in 3-manifolds. Specifically we consider the lifts to the universal cover of the stable and unstable foliations and analyze the leaf spaces of these foliations. We completely determine the structure of the non Hausdorff points in these leaf spaces. There are many consequences: (1) when the leaf spaces are non Hausdorff, there are closed orbits in the manifold which are freely homotopic, (2) suspension Anosov flows are, up to topological conjugacy, the only Anosov flows without free homotopies between closed orbits, (3) when there are infinitely many stable leaves (in the universal cover) which are non separated from each other, then we produce a torus in the manifold which is transverse to the Anosov flow and therefore is incompressible, (4) we produce non Hausdorff examples in hyperbolic manifolds and derive important properties of the limit sets of the stable/unstable leaves in the universal cover.

*Keywords: *Anosov flows, stable and unstable foliations, closed orbits, homotopic behavior of closed orbits, transversal submanifolds, non Hausdorff manifolds, hyperbolic manifolds and limit sets.

Fenley S.: The structure of branching in Anosov flows of 3-manifolds. *Comment. Math. Helv.* 73 (1998), 259-297. doi: 10.1007/s000140050055