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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Rational maps with real multipliers


Authors: Alexandre Eremenko and Sebastian van Strien
Journal: Trans. Amer. Math. Soc. 363 (2011), 6453-6463
MSC (2010): Primary 37F10, 30D05
Published electronically: July 25, 2011
MathSciNet review: 2833563
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Abstract: Let $ f$ be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever $ J(f)$ belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle.


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Additional Information

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

Sebastian van Strien
Affiliation: Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: strien@maths.warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05308-0
PII: S 0002-9947(2011)05308-0
Received by editor(s): November 13, 2008
Received by editor(s) in revised form: December 15, 2009
Published electronically: July 25, 2011
Additional Notes: The first author was supported by NSF grant DMS-0555279.
The second author was supported by a Royal Society Leverhulme Trust Senior Research Fellowship.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.