{"title":"Julia Components of Transcendental Entire Functions with Multiply-Connected Wandering Domains","authors":"","doi":"10.1007/s40315-024-00521-y","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We investigate some topological properties of Julia components, that is, connected components of the Julia set, of a transcendental entire function <em>f</em> with a multiply-connected wandering domain. If <em>C</em> is a Julia component with a bounded orbit, then we show that there exists a polynomial <em>P</em> such that <em>C</em> is homeomorphic to a Julia component of the Julia set of <em>P</em>. Furthermore if <em>C</em> is wandering, then <em>C</em> is a buried singleton component. Also we show that under some dynamical conditions, every such <em>C</em> is full and a buried component. The key for our proof is to show that some iterate of <em>f</em> can be regarded as a polynomial-like map on a suitable arbitrarily large bounded topological disk. As an application of this result, we show that a transcendental entire function having a wandering domain with a bounded orbit cannot have multiply-connected wandering domains.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40315-024-00521-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate some topological properties of Julia components, that is, connected components of the Julia set, of a transcendental entire function f with a multiply-connected wandering domain. If C is a Julia component with a bounded orbit, then we show that there exists a polynomial P such that C is homeomorphic to a Julia component of the Julia set of P. Furthermore if C is wandering, then C is a buried singleton component. Also we show that under some dynamical conditions, every such C is full and a buried component. The key for our proof is to show that some iterate of f can be regarded as a polynomial-like map on a suitable arbitrarily large bounded topological disk. As an application of this result, we show that a transcendental entire function having a wandering domain with a bounded orbit cannot have multiply-connected wandering domains.
摘要 我们研究了具有多重连接游走域的超越全函数 f 的 Julia 分量（即 Julia 集的连接分量）的一些拓扑性质。如果 C 是一个有界轨道的 Julia 分量，那么我们证明存在一个多项式 P，使得 C 与 P 的 Julia 集的 Julia 分量同构。我们还证明，在某些动力学条件下，每个这样的 C 都是完整的，并且是一个埋没的分量。我们证明的关键在于证明 f 的某些迭代可以被视为任意大的有界拓扑盘上的多项式类映射。作为这一结果的应用，我们证明了具有有界轨道徘徊域的超越全函数不可能具有多重连接的徘徊域。
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