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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 C9? 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' Riemannian metric on M has positive topological entropy.

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