TOPOLOGICAL-ENTROPY FOR GEODESIC-FLOWS ON FIBER-BUNDLES OVER RATIONALLY HYPERBOLIC MANIFOLDS

Let M be the total space of a fibre bundle with base a simply connecte d manifold whose loop space homology grows exponentially for a given c oefficient field. Then we show that for any C-infinity Riemannian metr ic g on M, the topological entropy of the geodesic Row of g is positiv e. It follows then, that there exist closed manifolds M with arbitrary fundamental group, for which the geodesic Row of any C-infinity Riema nnian metric on M has positive topological entropy.