Continuity of filled-in Julia sets and the closing lemma

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.

A note on non-converging Julia sets

“…The convergence of Julia sets is studied in [Kri88,Kri89] for rational functions and in [Kra93] for the polynomials P m,λ (z) := λ (1 + z/m) m converging locally uniformly to E λ (z) := λe z , the example investigated in [DGH86]. Theorem 1 in [Kra93] states that the Julia sets J (P d,λ ) converge to J (E λ ) for values of λ such that, (i) the Fatou set F (E λ ) consists of the basin of an attracting periodic orbit, or (ii) F (E λ ) is empty.…”

“…Theorem 1 in [Kra93] states that the Julia sets J (P d,λ ) converge to J (E λ ) for values of λ such that, (i) the Fatou set F (E λ ) consists of the basin of an attracting periodic orbit, or (ii) F (E λ ) is empty. The method of proof in [Kra93] does not depend on the functions P d,λ and E λ and the result has been generalized to the class of meromorphic functions in [KK95a,Kri95]. Independently, Kisaka studied its generalization to the class of entire functions in [Kis95].…”

“…Independently, Kisaka studied its generalization to the class of entire functions in [Kis95]. KK95a,or Theorem 5.3,Kri95]. If F (g) is the union of basins of attracting periodic orbits and ∞ ∈ J (g), then J (g m ) converges to J (g) in the Hausdorff metric.…”

Carathéodory Convergence of Immediate Basins of Attraction to a Siegel Disk

“…We mention the work of A. Douady [5], who investigates the mapping f → J (f, C) from the class of polynomials of fixed degree to the set of nonempty plane compacta equipped with the Hausdorff metric d H (X, Y ) := max{∂(X, Y ), ∂(Y, X)}, ∂(X, Y ) := sup x∈X dist(x, Y ). We also mention subsequent papers [6,7,8,9,10,11] dealing with other classes of functions. Continuity of Julia sets is closely related to behaviour of connected components of the Fatou set containing periodic points.…”

Approximation of Baker domains and convergence of Julia sets.

“…The cases not considered until now are that g λ has an attracting basin A 0 containing the unique free critical point c 0 or a Siegel disc, in which case an irrational fixed point belongs to the Fatou set. Kriete consider in [47] a family of meromorphic functions f n converging uniformly on C to a meromorphic function f . He proved the Hausdorff convergence of the filled Julia set, when f fulfils certain conditions, that is, when f has no wandering domains, Baker domains or rationally indifferent cycles.…”