Recurrent critical points and typical limit sets for conformal measures

For a rational f :Ĉ→Ĉ with a conformal measure μ we show that if there is a subset of the Julia set J(f) of positive μ-measure whose points are not eventual preimages of critical or parabolic points and have limit sets not contained in the union of the limit sets of recurrent critical points, then μ is non-atomic, μ(J(f))=1, ω(x)=J(f) for μ-a.e. point x∈J(f) and f is conservative, ergodic and exact. The proof uses a version of the Lebesgue Density Theorem valid for Borel measures and conformal balls.