On fundamental loops and the fast escaping set

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.

Cited by in corpus (6)

• Maximally and non-maximally fast escaping points of transcendental entire functions
• The size and topology of quasi-Fatou components of quasiregular maps
• Hollow quasi-Fatou components of quasiregular maps
• Multiply connected wandering domains of meromorphic functions: internal dynamics and connectivity
• Iterates of Meromorphic Functions on Escaping Fatou Components
• Area of the complement of the fast escaping sets of a family of entire functions