On the connectivity of Julia sets of transcendental entire functions
On the connectivity of Julia sets of transcendental entire functions
复制标题
论 Julia 超越整体函数集的连通性
DOI：
--
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发表时间：
1998
影响因子：
0.9
通讯作者：
M. Kisaka
中科院分区：
文献类型：
--
作者：
M. Kisaka
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.
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.