Cutpoints of Invariant Subcontinua of Polynomial Julia Sets

Q3 Mathematics Arnold Mathematical Journal Pub Date : 2021-08-16 DOI:10.1007/s40598-021-00186-8

Alexander Blokh, Lex Oversteegen, Vladlen Timorin

引用次数: 3

Abstract

We prove fixed point results for branched covering maps f of the plane. For complex polynomials P with Julia set \(J_{P}\) these imply that periodic cutpoints of some invariant subcontinua of \(J_{P}\) are also cutpoints of \(J_{P}\). We deduce that, under certain assumptions on invariant subcontinua Q of \(J_{P}\), every Riemann ray to Q landing at a periodic repelling/parabolic point \(x\in Q\) is isotopic to a Riemann ray to \(J_{P}\) relative to Q.

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多项式Julia集不变次连续线的截点

我们证明了平面的分支覆盖映射f的不动点结果。对于具有Julia集的复多项式P（J_。我们推导出，在对\（J_｛P｝\）的不变子连续性Q的某些假设下，每一条到Q的黎曼射线都落在周期排斥/抛物点\。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Arnold Mathematical Journal Mathematics-Mathematics (all)

CiteScore

1.50

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0.00%

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期刊介绍： The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.

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