Geometric rigidity for class $\mathcal {S}$ of transcendental meromorphic functions whose Julia sets are Jordan curves
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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.References
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Additional Information
- Mariusz Urbański
- Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430
- Email: urbanski\@@unt.edu
- 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
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 3733-3739
- MSC (2000): Primary 30D05
- DOI: https://doi.org/10.1090/S0002-9939-09-09918-3
- MathSciNet review: 2529881