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Eremenko points and the structure of the escaping set

Rippon, Philip and Stallard, Gwyneth (2019). Eremenko points and the structure of the escaping set. Transactions of the American Mathematical Society (In Press).

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Abstract

Much recent work on the iterates of a transcendental entire function f has been motivated by Eremenko's conjecture that all the components of the escaping set I(f) are unbounded. We prove several general results about the topological structure of I(f) including the fact that if I(f) is disconnected, then it contains uncountably many pairwise disjoint unbounded continua, all of which are subsets of AR(f), the `core' of the fast escaping set. We also show that, for some R > 0, the set AR(f) is connected and has the structure of an infinite spider's web or it contains uncountably many unbounded connected Fσ sets. There are analogous results for the intersections of these sets with the Julia set when multiply connected wandering domains are not present, but very different results when such wandering domains are present. In proving these, we obtain the unexpected result that some types of multiply connected wandering domains have complementary components with no interior, indeed uncountably many.

Item Type: Journal Item
ISSN: 1088-6850
Project Funding Details:
Funded Project NameProject IDFunding Body
Bakers Conjecture and Eremenko's Conjecture: New Directions (XM-12-066-GS)EP/K031163/1EPSRC (Engineering and Physical Sciences Research Council)
Keywords: escaping set; Cantor bouquet; spider's web; Wiman-Valiron; fast escaping set; multiply connected wandering domain
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 55614
Depositing User: Philip Rippon
Date Deposited: 28 Jun 2018 15:18
Last Modified: 28 Jun 2018 15:18
URI: http://oro.open.ac.uk/id/eprint/55614
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