We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{ 1+\epsilon}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^\infty$ surface has at least const $\times(e^{ hT}/T)$ simple closed orbits of period less than $T$, whenever the topological entropy $h$ is positive -- and without further assumptions on the curvature.
Categories: math.DS math.DG