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

TL;DR

This work develops a cohesive combinatorial framework for the dynamics of post-singularly finite transcendental entire functions by introducing filaments as generalized escaping sets and constructing dynamic partitions that transfer planar landing information to the space of external addresses via itineraries. It proves that periodic and preperiodic filaments land in the extended plane in a way captured by these itineraries, and it shows the existence of a spider (and an invariant spider for some iterate) that parallels Hubbard trees for polynomials, enabling homotopy-based combinatorial models. By extending the complex plane to include iterated asymptotic tracts and by defining simple dynamic partitions, the authors establish a robust mechanism to classify landing behavior and to relate it to external-address dynamics, paving the way for a systematic combinatorial theory, including Hubbard trees, for psf entire maps.

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.

Figures (14)

• Figure 1: Sketch of a function with three transcendental singularities over three distinct asymptotic values.
• Figure 2: A sketch of a possible configuration of $D$ and $\alpha$.
• Figure 3: Sketch of $W_0$ and a fundamental domain $F$.
• Figure 4: Sketch of some fundamental tails of various levels. For a given address $\underline{s}$, the fundamental tails $\tau_n(\underline{s})$ might be nested or not. Fundamental tails of different levels might intersect, even if one is not the prefix of the other.
• Figure 5: Sketch of the unbounded component of a fundamental domain $F$.

Theorems & Definitions (108)

• Theorem A: Existence of spiders
• Theorem B: Landing equivalence
• Lemma 2.1: Coverings of punctured conformal disks
• proof
• Definition 2.2: Post-singulary finite (psf) entire functions
• Proposition 2.3: Periodic points and Fatou components of psf maps
• proof
• Lemma 2.4: Logarithmic singularities
• proof
• Proposition 2.5: Basic properties of the extension $\mathbb{C}_{{f}^{\mathsmaller{\mathsmaller{\infty}}}}$