Fast escaping points of entire functions

Abstract

Let f be a transcendental entire function and let A(f) denote the set of points that escape to infinity ‘as fast as possible’ under iteration. By writing A(f) as a countable union of closed sets, called ‘levels’ of A(f), we obtain a new understanding of the structure of this set. For example, we show that if U is a Fatou component in A(f), then ∂U⊂A(f) and this leads to significant new results and considerable improvements to existing results about A(f). In particular, we study functions for which A(f), and each of its levels, has the structure of an ‘infinite spider's web’. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.