  We investigate typical properties of nonexpansive mappings on unbounded
complete hyperbolic metric spaces. For two families of metrics of uniform
convergence on bounded sets, we show that the typical nonexpansive mapping is a
Rakotch contraction on every bounded subset and that there is a bounded set
which is mapped into itself by this mapping. In particular, we obtain that the
typical nonexpansive mapping in this setting has a unique fixed point which can
be reached by iterating the mapping. Nevertheless, it turns out that the
typical mapping is not a Rakotch contraction on the whole space and that it has
the maximal possible Lipschitz constant of one on a residual subset of its
domain. By typical we mean that the complement of the set of mappings with this
property is $\sigma$-$\phi$-porous, that is, small in a metric sense. For a
metric of pointwise convergence, we show that the set of Rakotch contractions
is meagre.
