On density of horospheres in dynamical laminations

Sullivan's dictionary relates two domains of complex dynamics: Kleinian groups and rational iterations on the Riemann sphere. In 1997 M. Lyubich and Y. Minsky have extended the Sullivan's dictionary by constructing an analogue of the hyperbolic manifold of a Kleinian group: the so-called quotient hyperbolic lamination associated to a rational function. This is an abstract topological space constructed from the space of backward orbits of the rational function that carries a “foliation” (more precisely, lamination) by hyperbolic 3-manifolds (that may be singular). The hyperbolic leaves are dense, may be after deleting at most finite number of isolated leaves. Each hyperbolic leaf is foliated by horospheres, which form the unstable foliation (horospheric lamination) for the leafwise vertical geodesic flow. We consider the total laminated space with isolated hyperbolic leaves deleted. We prove that the horospheric lamination is topologically transitive (and there are a lot of dense horospheres), if and only if the corresponding rational function does not belong to the following list of exceptions: powers, Chebyshev polynomials, Lattès examples. We show that the horospheric lamination is minimal, if the corresponding function does not belong to the same list of exceptions and is critically nonrecurrent without parabolics.