We consider a rational map f : Ĉ →Ĉ of the Riemann sphere with normalized Lebesgue measure μ and show that if there is a subset of the Julia set J(f) of positive μ-measure whose points have limit sets not contained in the union of the limit sets of recurrent critical points, then ω(x) = Ĉ = J(f) for μ-a.e. point x and f is conservative, ergodic and exact. © 1999 American Mathematical Society.