Abstract

Anosov flows on 3-manifolds are dynamical systems generalizing the behavior of geodesic flow on the unit tangent bundle of a hyperbolic surface.  Like geodesic flows, they come with two transverse, invariant 2-dimensional foliations which meet along the 1-dimensional foliation by orbits.  Because of various surgery techniques, there are many, many examples known, and their “topological” classification is an interesting and important problem both in low-dimensional geometric topology and dynamics.  I will describe some of this beautiful framework, and then some joint work with T. Barthelmé and S. Frankel (and more recently also with S. Fenley and C. Bonatti) on the classification problem.  Notably, with Barthelmé and Frankel, we have a complete invariant for transitive flows, which in many cases reduces simply to knowing the free homotopy classes represented by periodic orbits.