We consider a sequence of entire functions fg m g converging to a limit function g locally uniformly on C . In Kis95] it is claimed that, if the Fatou set F (g) of the limit function is the union of the basins of attracting periodic orbits, then the Julia sets J(g m ) converge to the Julia set J(g) in the Hausdor metric. We show that this is not true in general.

In this paper we discuss the continuity of filled-in Julia sets of functions meromorphic in the complex plane, i.e. rational or transcendental functions, or polynomials. The Main Theorem is: The filled-in Julia set depends continuously on the function provided the function in question has no Baker domain, wandering domain or parabolic cycle (theorem 3.1). The proofs are based on homotopy arguments and do not require any assumption on the number of singular values, actually, they simultaneously work for rational and transcendental functions. By examples we show the Main Theorem to be sharp. In order to illustrate the usage of filled-in Julia sets, applications to (relaxed) Newton's method are described. Using the continuity result a closing lemma for polynomials and entire transcendental functions is proven.

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Abstract. We study families G(λ, ·) of entire functions that are approximated by a sequence of families G n (λ, ·) of entire functions, where λ ∈ C is a parameter. In order to control the dynamics, the families are assumed to be of the same constant finite type. In this setting we prove the convergence of the hyperbolic components in parameter space as kernels in the sense of Carathéodory. IntroductionIn iteration theory, founded by Fatou and Julia [11,13], there has been much progress on the iteration of rational functions in the last few decades; for an overview see [4,21]. However, even though Fatou studied the iteration of entire transcendental functions, the transcendental case has only recently received much attention; see the expositions [1, 3, 8, 9]. The theory develops along the lines of the rational case, but often other and generally more complicated methods of proof need to be used. There is also a variety of phenomena that do not occur in the rational case. One question is: which results carry over from the rational case and which do not?An interesting approach to this question was suggested in [6] and is illustrated in [5, 15, 16]. The polynomials P n (λ, z) = λ(1+(z/n)) n converge uniformly on compact sets of the complex plane C to the exponentials E(λ, z) = λe z as n tends to infinity. In [6] a combinatorial description is given how external rays to the connectedness loci of the P n (λ, ·) converge to certain 'hairs' in the parameter plane for E(λ, ·). Furthermore, the pointwise convergence of hyperbolic components is shown in [5,6,15]. In the dynamical plane the convergence of Julia sets with respect to the Hausdorff metric is shown in [15] for the above families for suitable values of the parameter λ. The convergence of Julia sets was obtained in [7] for polynomials of constant degree, in [19] for rational functions, and has now been generalized to larger classes of functions; see [14,16,17,20].In this paper we are interested in convergence of hyperbolic components in parameter space for entire functions of the same constant finite type (for the exact definition see §3).

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