Hausdor Convergence of Julia Sets

Consider a sequence fg d g converging uniformly on compact sets to g, where g and g d are meromorphic functions on C. We show that the Julia sets J(g d) converge to the Julia set J(g) in the Hausdorr metric, if the Fatou set F (g) is the union of basins of attraction of attracting periodic orbits and 1 2 J(g). This result is discussed for families of nite type depending on a parameter, which is illustrated with the polynomials (1 + z d) d converging to e z .