  In this article we consider asymptotically harmonic manifolds which are
simply connected complete Riemannian manifolds without conjugate points such
that all horospheres have the same constant mean curvature $h$. We prove the
following equivalences for asymptotically harmonic manifolds $X$ under the
additional assumption that their curvature tensor together with its covariant
derivative are uniformly bounded: (a) $X$ has rank one; (b) $X$ has Anosov
geodesic flow; (c) $X$ is Gromov hyperbolic; (d) $X$ has purely exponential
volume growth with volume entropy equals $h$. This generalizes earlier results
by G. Knieper for noncompact harmonic manifolds and by A. Zimmer for
asymptotically harmonic manifolds admitting compact quotients.
