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Perturbations of graphs for Newton maps I: bounded hyperbolic components

Part of: Complex dynamical systems

Published online by Cambridge University Press:  31 August 2022

YAN GAO  and
HONGMING NIE
YAN GAO
Affiliation:
College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, China (e-mail: gyan@szu.edu.cn)
HONGMING NIE*
Affiliation:
Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794, USA
*
e-mail: hongming.nie@stonybrook.edu
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Abstract

We consider graphs consisting of finitely many internal rays for degenerating Newton maps and state a convergence result. As an application, we prove that a hyperbolic component in the moduli space of quartic Newton maps is bounded if and only if every element has degree $2$ on the immediate basin of each root. This provides the first complete description of bounded hyperbolic components in a complex two-dimensional moduli space.

Keywords

Newton mapsinternal rayshyperbolic componentsNewton graphs

MSC classification

Primary: 37F20: Combinatorics and topology
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 9 , September 2023 , pp. 2997 - 3025
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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