Brushing the hairs of transcendental entire functions (Q411811)

\textit{J. M. Aarts} and \textit{L. G. Oversteegen} [Trans. Am. Math. Soc. 338, No. 2, 897--918 (1993; Zbl 0809.54034)] showed that the Julia sets of \(\lambda e^z\), where \(0<\lambda<1/e\), and \(\lambda \sin z\), where \(0<\lambda <1\), are all homeomorphic to a certain subset of the plane which they called a straight brush. In fact, they showed that these Julia sets are even ambiently homeomorphic to a straight brush, meaning that the homeomorphism between the Julia set and the straight brush extends to a homeomorphism of the plane.NEWLINENEWLINEHere it is shown that this result also holds for the Julia sets of entire functions in the Eremenko-Lyubich class which are of finite order of growth and for which the Fatou set consists of a single attracting basin containing the closure of the set of singularities of the inverse. Earlier results, partially by the same authors, showed that for such functions the Julia set consists of disjoint arcs to infinity. If the assumption about the Fatou set is dropped, the Julia set still contains a Cantor bouquet containing all points staying large under iteration.