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Dynamics of Newton maps

Part of: Complex dynamical systems

Published online by Cambridge University Press:  15 February 2022

XIAOGUANG WANG ,
YONGCHENG YIN  and
JINSONG ZENG
XIAOGUANG WANG
Affiliation:
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, P. R. China (e-mail: wxg688@163.com, yin@zju.edu.cn)
YONGCHENG YIN
Affiliation:
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, P. R. China (e-mail: wxg688@163.com, yin@zju.edu.cn)
JINSONG ZENG*
Affiliation:
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China
*
e-mail: jinsongzeng@163.com
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Abstract

In this paper, we study the dynamics of the Newton maps for arbitrary polynomials. Let p be an arbitrary polynomial with at least three distinct roots, and f be its Newton map. It is shown that the boundary $\partial B$ of any immediate root basin B of f is locally connected. Moreover, $\partial B$ is a Jordan curve if and only if $\mathrm {deg}(f|_B)=2$. This implies that the boundaries of all components of root basins, for the Newton maps for all polynomials, from the viewpoint of topology, are tame.

Keywords

Newton mapspuzzleslocal connectivity

MSC classification

Primary: 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 3 , March 2023 , pp. 1035 - 1080
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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