Dynamics of Newton maps

TL;DR

This paper investigates the topological properties of Newton maps for arbitrary polynomials, proving local connectivity of basin boundaries and characterizing when these boundaries are Jordan curves.

Contribution

It establishes the local connectivity of basin boundaries for Newton maps and characterizes when these boundaries are Jordan curves, extending understanding of their topological structure.

Findings

Boundaries of immediate root basins are locally connected.

Boundaries are Jordan curves if and only if the degree of the restriction is 2.

All basin boundaries are topologically tame.

Abstract

In this paper, we study the dynamics of 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 ${\rm deg}(f|_B)=2$. This implies that the boundaries of all components of root basins, for all polynomials' Newton maps, from the viewpoint of topology, are tame.