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

 

Hausdorff dimensions of escaping sets of transcendental entire functions


Authors: Lasse Rempe and Gwyneth M. Stallard
Journal: Proc. Amer. Math. Soc. 138 (2010), 1657-1665
MSC (2000): Primary 37F10; Secondary 37F35, 30D05
Published electronically: December 18, 2009
MathSciNet review: 2587450
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Abstract: Suppose that $ f$ and $ g$ are transcendental entire functions, each with a bounded set of singular values, and that $ g\circ\phi = \psi\circ f$, where $ \phi,\psi:\mathbb{C}\to\mathbb{C}$ are affine. We show that the escaping sets of $ f$ and $ g$ have the same Hausdorff dimension.

Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one.


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Additional Information

Lasse Rempe
Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom
Email: l.rempe@liverpool.ac.uk

Gwyneth 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/S0002-9939-09-10104-1
PII: S 0002-9939(09)10104-1
Received by editor(s): April 20, 2009
Published electronically: December 18, 2009
Additional Notes: Both authors are supported by the European CODY network. The first author is supported by EPSRC fellowship EP/E052851/1.
Communicated by: Mario Bonk
Article copyright: © Copyright 2009 American Mathematical Society