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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Boundaries of escaping Fatou components


Authors: P. J. Rippon and G. M. Stallard
Journal: Proc. Amer. Math. Soc. 139 (2011), 2807-2820
MSC (2010): Primary 37F10; Secondary 30D05
Published electronically: March 2, 2011
MathSciNet review: 2801622
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Abstract: Let $ f$ be a transcendental entire function and $ U$ be a Fatou component of $ f$. We show that if $ U$ is an escaping wandering domain of $ f$, then most boundary points of $ U$ (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of $ U$ are escaping, then $ U$ is an escaping Fatou component. Some applications of these results are given; for example, if $ I(f)$ is the escaping set of $ f$, then $ I(f)\cup\{\infty\}$ is connected.


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Additional Information

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

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

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10842-6
PII: S 0002-9939(2011)10842-6
Received by editor(s): July 23, 2010
Published electronically: March 2, 2011
Additional Notes: Both authors were supported by EPSRC grant EP/H006591/1.
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.