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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Eremenko points and the structure of the escaping set


Authors: P. J. Rippon and G. M. Stallard
Journal: Trans. Amer. Math. Soc. 372 (2019), 3083-3111
MSC (2010): Primary 30D05, 37F10
DOI: https://doi.org/10.1090/tran/7673
Published electronically: June 10, 2019
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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 the fast escaping set. We give analogous results for the intersection of $ I(f)$ with the Julia set when multiply connected wandering domains are not present, and show that completely different results hold 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.


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Additional Information

P. J. Rippon
Affiliation: School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
Email: phil.rippon@open.ac.uk

G. M. Stallard
Affiliation: School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
Email: gwyneth.stallard@open.ac.uk

DOI: https://doi.org/10.1090/tran/7673
Keywords: Escaping set, Cantor bouquet, spider's web, Wiman--Valiron, fast escaping set, multiply connected wandering domain
Received by editor(s): September 15, 2017
Received by editor(s) in revised form: March 12, 2018
Published electronically: June 10, 2019
Additional Notes: Both authors were supported by EPSRC grant EP/K031163/1.
Article copyright: © Copyright 2019 American Mathematical Society