Measured solenoidal Riemann surfaces and holomorphic dynamics

We study, from a measure theoretic point of view, the lamination structure on the inverse limit space <(lim)under left arrow>((C) over bar, f) for an arbitrary rational map f on the sphere (C) over bar. It turns out that there is an ergodic holomorphic foliated dynamical object L, namely a self mapping of a measured solenoidal Riemann surface, which continuously injects into the inverse limit space, with full image and with leaves conformally isomorphic to the complex plane C.