Perturbations of graphs for Newton maps

TL;DR

This paper investigates how certain graph structures associated with degenerating Newton maps converge and applies these results to analyze the boundedness of hyperbolic components in the moduli space of quartic Newton maps.

Contribution

It provides a sufficient condition for the convergence of graphs in degenerating Newton maps and characterizes bounded hyperbolic components based on local degrees.

Findings

Convergence of internal ray graphs is guaranteed under specific conditions.

Bounded hyperbolic components correspond to maps with degree 2 on immediate basins.

The study links graph convergence to the geometric structure of Newton maps.

Abstract

We study the convergence of graphs consisting of finitely many internal rays for degenerating Newton maps. We state a sufficient condition to guarantee the convergence. As an application, we investigate the boundedness of hyperbolic components in the moduli space of quartic Newton maps. We prove that such a hyperbolic component is bounded if and only if every element has degree $2$ on the immediate basin of each root.