In this paper we study the computability of the stable and unstable manifolds
of a hyperbolic equilibrium point. These manifolds are the essential feature
which characterizes a hyperbolic system. We show that (i) locally these
manifolds can be computed, but (ii) globally they cannot (though we prove they
are semi-computable). We also show that Smale's horseshoe, the first example of
a hyperbolic invariant set which is neither an equilibrium point nor a periodic
orbit, is computable
