  We suggest a way to associate to a rational map of the Riemann sphere a three
dimensional object called a hyperbolic orbifold 3-lamination. The relation of
this object to the map is analogous to the relation of a hyperbolic 3-manifold
to a Kleinian group. In order to construct the 3-lamination we analyze the
natural extension of a rational map and the complex affine structure on the
canonical 2-dimensional leaf space contained in it. In this paper the
construction is carried out in full for post-critically finite maps. We show
that the corresponding laminations have a compact convex core. As a first
application we give a three-dimensional proof of Thurston's rigidity for
post-critically finite mappings, via the "lamination extension" of the proofs
of the Mostow and Marden rigidity and isomorphism theorems for hyperbolic
3-manifolds. An Ahlfors-type argument for zero measure of the Julia set is
applied along the way. This approach also provides a new point of view on the
Lattes deformable examples.
