In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions and Kleinian groups (both acting on 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: hyperbolic laminations. Appropriate surgery on the backward orbit space yields an abstract topological space foliated by hyperbolic 3- manifolds (with singularities). The nonbijective action of the rational function lifts up to a bijective action on the latter space by isometries of leaves. The quotient of the latter action is a nice space (called the quotient hyperbolic lamination). It is foliated by hyperbolic 3- manifolds (may be with singularities) with marked point "infinity" on the boundaries of their covering hyperbolic spaces. The horospheres passing through infinity (i.e., the horizontal planes of the hyperbolic 3- space in the half-space model) induce a foliation by surfaces in the quotient lamination space (called the horospheric lamination).
Recent studying the hyperbolic 3-manifolds associated to Kleinian groups resulted in solutions of all big problems in the theory. There is a hope that studying the hyperbolic laminations associated to rational functions would imply important dynamical corollaries.
One of the main results of the talk says that the horospheric lamination is topologically transitive, provided that the rational function under consideration does not belong to an explicit list of exceptions (for which this is not true).