Rippon, Philip and Stallard, Gwyneth
Fast escaping points of entire functions.
Proceedings of the London Mathematical Society, 105(4) pp. 787–820.
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Let be a transcendental entire function and let denote the set of points that escape to infinity `as fast as possible’ under iteration. By writing as a countable union of closed sets, called `levels’ of , we obtain a new understanding of the structure of this set. For example, we show that if is a Fatou component in , then and this leads to significant new results and considerable improvements to existing results about . In particular, we study functions for which , 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.
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