Mandelbrot set for fractal n-gons and zeros of power series

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-04-16 DOI:10.1016/j.topol.2024.108918

Yuto Nakajima

引用次数: 0

Abstract

We give a framework to study the connectedness of the set of zeros of power series with coefficients in a finite subset $G \subset C$ . We prove that the set of zeros in the unit disk is connected and locally connected if some graph on the set G of coefficients is connected. Furthermore, we apply this result to the study of the Mandelbrot set $M_{n}$ for fractal n-gons. We prove that $M_{n}$ is connected and locally connected for any n.

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分形正圆和幂级数零点的曼德布罗特集

我们给出了一个研究系数在有限子集 G⊂C 中的幂级数零点集连通性的框架。我们证明，如果系数集 G 上的某个图是连通的，那么单位盘中的零点集就是连通的，而且是局部连通的。此外，我们还将这一结果应用于研究分形 n 形的曼德尔布罗特集 Mn。我们证明，对于任意 n，Mn 都是连通和局部连通的。

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

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.