A Combinatorial classification of postcritically fixed Newton maps

TL;DR

This paper introduces a combinatorial framework for classifying postcritically fixed Newton maps, establishing foundational results that could extend to broader classes of Newton maps in complex dynamics.

Contribution

It provides the first combinatorial classification for postcritically fixed Newton maps and proves connectivity properties of their basins of attraction.

Findings

Connected components of basins can be linked to infinity via finite chains.

Framework sets the stage for classifying more general Newton maps.

Advances understanding of the structure of Newton map dynamics.

Abstract

We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to $\infty$ through a finite chain of such components.