Combinatorics of criniferous entire maps with escaping critical values

TL;DR

This paper develops combinatorial tools to describe the escaping set of criniferous entire functions without asymptotic values, especially when escaping critical values cause splitting, advancing understanding of their complex dynamics.

Contribution

It introduces a complete combinatorial description of the escaping set for criniferous functions without asymptotic values, accounting for critical value splitting phenomena.

Findings

Provides a detailed combinatorial framework for escaping sets.

Reflects the splitting at critical points in the structure.

Lays groundwork for analyzing topological dynamics of such functions.

Abstract

A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at (preimages of) critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its Julia set. In particular, our description precisely reflects the splitting phenomenon. This combinatorial structure provides the foundation for further study of this class of functions. For example, we use these results in [arXiv:1905.03778] to give the first full description of the topological dynamics of a class of transcendental entire maps with unbounded postsingular set.