Contact Anosov flows on hyperbolic 3–manifolds

Abstract

Geodesic flows of Riemannian or Finsler manifolds have been the only known contact Anosov flows. We show that even in dimension 3 the world of contact Anosov flows is vastly larger via a surgery construction near an $E$–transverse Legendrian link that encompasses both the Handel–Thurston and Goodman surgeries and that produces flows not topologically orbit equivalent to any algebraic flow. This includes examples on many hyperbolic 3–manifolds, any of which have remarkable dynamical and geometric properties.

To the latter end we include a proof of a folklore theorem from 3–manifold topology: In the unit tangent bundle of a hyperbolic surface, the complement of a knot that projects to a filling geodesic is a hyperbolic 3–manifold.