The classification of polynomial basins of infinity

TL;DR

This paper introduces a new combinatorial invariant called the pictograph for classifying the dynamics of complex polynomials on their basins of infinity, providing a complete description for polynomials with all critical points escaping.

Contribution

It synthesizes existing tools into the pictograph invariant and offers algorithms for constructing and counting conjugacy classes based on this invariant.

Findings

Complete description of conjugacy classes for polynomials with escaping critical points

Algorithm for constructing abstract pictographs

Inductive method for counting conjugacy classes

Abstract

We consider the problem of classifying the dynamics of complex polynomials $f: \mathbb{C} \to \mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools --- tableaux, trees, and laminations --- into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to infinity, we obtain a complete description of the set of topological conjugacy classes. We give an algorithm for constructing abstract pictographs, and we provide an inductive algorithm for counting topological conjugacy classes with a given pictograph.