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

 

Geometric rigidity for class $ \mathcal{S}$ of transcendental meromorphic functions whose Julia sets are Jordan curves


Author: Mariusz Urbanski
Journal: Proc. Amer. Math. Soc. 137 (2009), 3733-3739
MSC (2000): Primary 30D05
Published electronically: May 28, 2009
MathSciNet review: 2529881
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Abstract: We consider any transcendental meromorphic function $ f$ of Class $ \mathcal{S}$ whose Julia set is a Jordan curve. We show that the Julia set of $ f$ either is an extended straight line or has Hausdorff dimension strictly greater than $ 1$. The proof uses conformal iterated function systems and extends many earlier results of this type.


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Additional Information

Mariusz Urbanski
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430
Email: urbanski\@@unt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09918-3
PII: S 0002-9939(09)09918-3
Keywords: Holomorphic dynamics, Hausdorff dimension, meromorphic functions
Received by editor(s): July 8, 2008
Received by editor(s) in revised form: February 16, 2009
Published electronically: May 28, 2009
Additional Notes: The author’s research was supported in part by NSF grant DMS 0700831. Part of the work was done while the author was visiting the Max Planck Institute in Bonn, Germany. He wishes to thank the institute for its support.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2009 American Mathematical Society