Dynamics of Newton maps

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

47 pages, 18 figures. Writing polished, results unchanged, more figures added