A sharp growth condition for a fast escaping spider¡¯s web

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• 作者：P.J. Rippon ; G.M. Stallard
• 关键词：primary ; 37F10 ; secondary ; 30D05
• 刊名：Advances in Mathematics
• 出版年：2013
• 出版时间：10 September, 2013
• 年：2013
• 卷：244
• 期：Complete
• 页码：337-353
• 全文大小：205 K

文摘

We show that the fast escaping set of a transcendental entire function has a structure known as a spider¡¯s web whenever the maximum modulus of grows below a certain rate. The proof uses a new local version of the 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.