# Eremenko points and the structure of the escaping set

## TL;DR

This paper investigates the structure of the escaping set of transcendental entire functions, showing that disconnectedness leads to uncountably many unbounded components, and characterizes the core fast escaping set as either connected or highly fragmented.

## Contribution

It provides new structural insights into the escaping set and the fast escaping set, including conditions for connectedness and the existence of unbounded components, and explores the impact of wandering domains.

## Key findings

- If the escaping set is disconnected, then its complement in any disk meeting the Julia set has uncountably many unbounded components.
- The core fast escaping set is either connected and forms an infinite spider's web or has uncountably many unbounded components.
- Multiply connected wandering domains can have uncountably many complementary components with no interior.

## Abstract

Much recent work on the iterates of a transcendental entire function $f$ has been motivated by Eremenko's conjecture that all the components of the escaping set $I(f)$ are unbounded. Here we show that if $I(f)$ is disconnected, then the set $I(f)\setminus D$ has uncountably many unbounded components for any open disc $D$ that meets the Julia set of $f$. For the set $A_R(f)$, which is the `core' of the fast escaping set, we prove the much stronger result that for some $R>0$ either $A_R(f)$ is connected and has the structure of an infinite spider's web or it has uncountably many components each of which is unbounded. There are analogous results for the intersections of these sets with the Julia set when no multiply connected wandering domains are present, but strikingly different results when they are present. In proving these, we obtain the unexpected result that multiply connected wandering domains can have complementary components with no interior, indeed uncountably many.