  In 1985 D.Sullivan had introduced a dictionary between two domains of complex
dynamics: iterations of rational functions on the Riemann sphere and Kleinian
groups. The latters are discrete subgroups of the group of conformal
automorphisms of the Riemann sphere. This dictionary motivated many remarkable
results in both domains, starting from the famous Sullivan's no wandering
domain theorem in the theory of iterations of rational functions.
  One of the principal objects used in the study of Kleinian groups is the
hyperbolic 3- manifold associated to a Kleinian group, which is the quotient of
its lifted action to the hyperbolic 3- space. M.Lyubich and Y.Minsky have
suggested to extend Sullivan's dictionary by providing an analogous
construction for iterations of rational functions. Namely, they have
constructed a lamination by three-dimensional manifolds equipped with a
continuous family with hyperbolic metrics on them (may be with singularities).
The action of the rational mapping on the sphere lifts naturally up to
homeomorphic action on the hyperbolic lamination that is isometric along the
leaves. The action on the hyperbolic lamination admits a well-defined quotient
called {\it the quotient hyperbolic lamination}.
  We study the arrangement of the horospheres in the quotient hyperbolic
lamination. The main result says that if a rational function does not belong to
a small list of exceptions (powers, Chebyshev and Latt\`es), then there are
many dense horospheres, i.e., the horospheric lamination is
topologically-transitive. We show that for "many" rational functions
(hyperbolic or critically-nonrecurrent nonparabolic) the quotient horospheric
lamination is minimal: each horosphere is dense.
