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Meromorphic functions with linearly distributed values and Julia sets of rational functions
Author(s):
Walter
Bergweiler;
Alexandre
Eremenko
Journal:
Proc. Amer. Math. Soc.
137
(2009),
2329-2333.
MSC (2000):
Primary 30D35
Posted:
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.
References:
-
- 1.
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- 2.
- A. Eremenko and S. van Strien, Rational functions with real multipliers, preprint, arXiv:0810.2260.
- 3.
- P. Fatou, Sur les équations fonctionnelles (Troisième Mémoire), Bull. Soc. Math. France, 48 (1920) 208-314. MR 1504797
- 4.
- A. Goldberg and I. Ostrovskii, Distribution of values of meromorphic functions (Russian). Nauka, Moscow, 1970. English translation: Amer. Math. Soc., Providence, RI, 2008. MR 0280720 (43:6439)
- 5.
- D. Hamilton, Length of Julia curves, Pacific J. Math., 169 (1995) 75-93. MR 1346247 (96m:30038)
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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:
10.1090/S0002-9939-08-09788-8
PII:
S 0002-9939(08)09788-8
Received by editor(s):
September 2, 2008
Posted:
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
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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