On fundamental loops and the fast escaping set

J. Lond. Math. Soc. (2) 88 (2013), no. 3, 716–736. Also available on the arXiv.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.