Hyperbolicity of complements of orbits in Anosov flows

We show that if an Anosov flow on a 3-dimensional manifold has orientable stable and unstable foliations, then the complement of any filling periodic orbit is a hyperbolic manifold. This generalizes the known case of the complement of a closed, filling geodesic orbit in the unit tangent bundle of a hyperbolic surface. Furthermore, we show that the orientability condition on invariant foliations is necessary, by constructing a counterexample in the absence of this property. In the case of the suspension flows we obtain that the complement of every collection of periodic orbits is hyperbolic, while for the geodesic flow (regardless of orientability of the invariant foliations) the complement of every filling and anannular collection of periodic orbits is hyperbolic.