A classification of postcritically finite Newton maps

TL;DR

This paper provides a complete combinatorial classification of postcritically finite Newton maps, a natural family of rational maps, using Thurston's theorems, advancing the understanding of their dynamical structure.

Contribution

It introduces a novel classification framework for postcritically finite Newton maps based on finite graphs and Thurston's theory, filling a gap in the dynamical classification of rational maps.

Findings

Complete combinatorial classification of postcritically finite Newton maps

Use of Thurston's characterization and rigidity theorem

Explicit conditions for the associated finite graphs

Abstract

The dynamical classification of rational maps is a central concern of holomorphic dynamics. Much progress has been made, especially on the classification of polynomials and some approachable one-parameter families of rational maps; the goal of finding a classification of general rational maps is so far elusive. Newton maps (rational maps that arise when applying Newton's method to a polynomial) form a most natural family to be studied from the dynamical perspective. Using Thurston's characterization and rigidity theorem, a complete combinatorial classification of postcritically finite Newton maps is given in terms of a finite connected graph satisfying certain explicit conditions.