Abstract
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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