Ergodic Theory and Dynamical Systems



On the connectivity of Julia sets of transcendental entire functions


MASASHI KISAKA a1
a1 Department of Mathematics, College of Integrated Arts and Sciences, Osaka Prefecture University, Gakuen-cho 1-1, Sakai 593, Japan (e-mail: kisaka@mi.cias.osakafu-u.ac.jp)

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 ${\Bbb C}$. 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 $\varphi : {\Bbb D} \longrightarrow U$ be a Riemann map of $U$ from a unit disk ${\Bbb D}$, then under some mild conditions we show that the set $\Theta_{\infty}$ of all angles where $\varphi$ admits the radial limit $\infty$ is dense in $\partial {\Bbb D}$ 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 $\Theta_{\infty}$ is dense in $\partial {\Bbb D}$ or at least its closure $\overline{\Theta_{\infty}}$ contains a certain perfect set, which means the boundary $\partial U$ has a very complicated structure. In all cases, this result leads to the disconnectivity of the Julia set $J_f$ in ${\Bbb C}$. If $U$ is a Baker domain on which $f$ is univalent, however, we shall show by giving an example that $\partial U$ can be a Jordan arc in ${\Bbb C}$, which has a rather simple structure, and, moreover, $J_f$ can be connected.

We also consider the connectivity of the set $J_f \cup \{ \infty \}$ in the Riemann sphere $\widehat{{\Bbb C}}$ and show that $J_f \cup \{ \infty \}$ is connected if and only if $f$ has no multiply-connected wandering domains.

(Received January 6 1996)
(Revised July 4 1996)