Hausdorff dimensions of escaping sets of transcendental entire functions
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- by Lasse Rempe and Gwyneth M. Stallard PDF
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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
- MR Author ID: 738017
- ORCID: 0000-0001-8032-8580
- 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
- MR Author ID: 292621
- Email: g.m.stallard@open.ac.uk
- 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
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 1657-1665
- MSC (2000): Primary 37F10; Secondary 37F35, 30D05
- DOI: https://doi.org/10.1090/S0002-9939-09-10104-1
- MathSciNet review: 2587450