A sharp growth condition for a fast escaping spider’s web

Rippon, P. J. and Stallard, G. M. (2013). A sharp growth condition for a fast escaping spider’s web. Advances in Mathematics, 244 pp. 337–353.

DOI: https://doi.org/10.1016/j.aim.2013.04.021

Abstract

We show that the fast escaping set A(f) of a transcendental entire function f has a structure known as a spider’s web whenever the maximum modulus of f grows below a certain rate. The proof uses a new local version of the cos πρ theorem, based on a comparatively unknown result of Beurling. We also give examples of entire functions for which the fast escaping set is not a spider’s web which show that this growth rate is sharp. These are the first examples for which the escaping set has a spider’s web structure but the fast escaping set does not. Our results give new insight into possible approaches to proving a conjecture of Baker, and also a conjecture of Eremenko.

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