Filament Pairs, Dynamic Partitions, and Spiders for Post-Singularly Finite Entire Functions

TL;DR

This paper extends the concept of dynamic rays to filaments in complex dynamics, providing a combinatorial framework for understanding their landing behavior and developing tools like spiders for classifying post-singularly finite entire functions.

Contribution

It offers a combinatorial description of filament landing relations and introduces invariant spiders for these functions, advancing the classification methods in complex dynamics.

Findings

Filaments share topological properties with dynamic rays.

A combinatorial model for filament landing relations is developed.

Every post-singularly finite entire function has an invariant spider.

Abstract

Filaments are a natural generalization of the well-known concept of dynamic rays in complex dynamics. In this article we investigate which periodic or preperiodic filaments land together for arbitrary post-singularly finite transcendental entire functions. Our first main result is a combinatorial description of the landing relation of filaments in terms of the dynamic partitions of the space of external addresses. One of the main difficulties deals with taming the more complicated topology of filaments. In the end, filaments possess all the topological properties of dynamic rays that are essential for the construction of dynamic partitions. The results of this paper are the foundation for the development of combinatorial models, in particular homotopy Hubbard trees, for arbitrary post-singularly finite transcendental entire functions. Our second mail result is that every postsingularly finite entire function has an iterated that possesses and invariant spider: spiders are, like homotopy Hubbard trees, an important tool for the combinatorial classification of postsingularly finite polynomials and presumably also for entire functions.