On the connectivity of Julia sets of transcendental entire functions

Abstract

We have two main purposes in this paper. One is to give some sufficient conditions for the Julia set of a transcendental entire function f to be connected or to be disconnected as a subset of the complex plane C. The other is to investigate the boundary of an unbounded periodic Fatou component U , which is known to be simply-connected. These are related as follows: let ϕ : D −→ U be a Riemann map of U from a unit disk D, then under some mild conditions we show that the set ∞ of all angles where ϕ admits the radial limit ∞ is dense in ∂D if U is an attracting basin, a parabolic basin or a Siegel disk. If U is a Baker domain on which f is not univalent, then ∞ is dense in ∂D or at least its closure ∞ contains a certain perfect set, which means the boundary ∂U has a very complicated structure. In all cases, this result leads to the disconnectivity of the Julia set J f in C. If U is a Baker domain on which f is univalent, however, we shall show by giving an example that ∂U can be a Jordan arc in C, which has a rather simple structure, and, moreover, J f can be connected. We also consider the connectivity of the set J f ∪ {∞} in the Riemann sphere C and show that J f ∪ {∞} is connected if and only if f has no multiply-connected wandering domains.

Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

“…Since J(f ) contains a spider's web, f has no multiply connected Fatou components. Therefore, all the components of J(f ) are unbounded (see [11]), and hence J(f ) is connected.…”

On questions of Fatou and Eremenko

“…For example, the function f (z) = 2 − log 2 + 2z − e z , discussed by Bergweiler in [4], has a Fatou component containing the point log 2 + 2πi, which is simply connected and wandering [4,Section 2]. This component can also be shown to be bounded (by finding a Jordan curve around log 2 + 2πi, which is mapped outside itself by f ; see [10]). …”

Irvine Noel Baker 1932–2001

“…Baker gave a complete description of all solutions of this equation, thus completing earlier work with Gross [21] for the case when f is entire. A Picard set is a set outside which every transcendental entire function takes all finite values with at most one exception infinitely often.…”