  We use Busemann functions to construct volume preserving mappings in an
asymptotically harmonic manifold. If the asymptotically harmonic manifold
satisfies the visibility condition, we construct mappings which preserve
distances in some directions. We also prove that some integrals on the
intersection of horospheres are independent of the differences between the
values of the corresponding Busemann functions and we establish an upper bound
of the volume of the intersection of two horospheres which is independent of
the difference between values of corresponding Busemann functions.
