Topological entropy for geodesic flows on fibre bundles over rationally hyperbolic manifolds

Abstract:

Let $M$ be the total space of a fibre bundle with base a simply connected manifold whose loop space homology grows exponentially for a given coefficient field. Then we show that for any $C^{\infty }$ Riemannian metric $g$ on $M$, the topological entropy of the geodesic flow of $g$ is positive. It follows then, that there exist closed manifolds $M$ with arbitrary fundamental group, for which the geodesic flow of any $C^{\infty }$ Riemannian metric on $M$ has positive topological entropy.