  Let M be a closed manifold whose based loop space is ``complicated''.
Examples are rationally hyperbolic manifolds and manifolds whose fundamental
group has exponential growth. We prove that the topological entropy of any Reeb
flow on the spherization of T*M is positive. Moreover, given q in M, for almost
every q' in M the number of Reeb orbits from the fiber over q to the fiber over
q' grows exponentially in time. If M is a surface of higher genus, we also
obtain exponential growth of the number of closed Reeb orbits.
