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

 

Meromorphic functions with linearly distributed values and Julia sets of rational functions


Authors: Walter Bergweiler and Alexandre Eremenko
Journal: Proc. Amer. Math. Soc. 137 (2009), 2329-2333
MSC (2000): Primary 30D35
Published electronically: December 22, 2008
MathSciNet review: 2495266
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Abstract: If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve, then it is contained in a circle.


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Additional Information

Walter Bergweiler
Affiliation: Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str.\ 4, D–24098 Kiel, Germany
Email: bergweiler@math.uni-kiel.de

Alexandre Eremenko
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: eremenko@math.purdue.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09788-8
PII: S 0002-9939(08)09788-8
Received by editor(s): September 2, 2008
Published electronically: December 22, 2008
Additional Notes: The first author was supported by the G.I.F., the German-Israeli Foundation for Scientific Research and Development, Grant G-809-234.6/2003; the EU Research Training Network CODY; and the ESF Research Networking Programme HCAA
The second author was supported by NSF grant DMS-0555279
Communicated by: Mario Bonk
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.