Abstract: The Bott-Samelson theorem states that a Riemannian manifold all of whose geodesics are closed must be rather special: either M is a circle or the integral cohomology ring of M is generated by one element. We show that this theorem is actually a "symplectic" theorem: it holds for Reeb flows on spherizations, which are the contact-dynamical generalizations of geodesic flows. The tools in the proof are a slow version of topological entropy, and Lagrangian Floer homology. This is joint work with Urs Frauenfelder and Clémence Labrousse.