On fundamental loops and the fast escaping set

TL;DR

This paper studies the structure of fundamental loops within the fast escaping set of transcendental entire functions, revealing diverse geometric properties and introducing a new escape rate function.

Contribution

It introduces a real-valued function to measure escape rates and analyzes the geometric structure of fundamental loops, especially for functions with multiply connected Fatou components.

Findings

Existence of fundamental loops that are approximately circular or highly distorted.

The escape rate function exhibits interesting mathematical properties.

Fundamental loops can vary significantly in geometry depending on the function.

Abstract

The fast escaping set, A(f), of a transcendental entire function f has begun to play a key role in transcendental dynamics. In many cases A(f) has the structure of a spider's web, which contains a sequence of fundamental loops. We investigate the structure of these fundamental loops for functions with a multiply connected Fatou component, and show that there exist transcendental entire functions for which some fundamental loops are analytic curves and approximately circles, while others are geometrically highly distorted. We do this by introducing a real-valued function which measures the rate of escape of points in A(f), and show that this function has a number of interesting properties.