Rational maps with real multipliers
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- by Alexandre Eremenko and Sebastian van Strien
- Trans. Amer. Math. Soc. 363 (2011), 6453-6463
- DOI: https://doi.org/10.1090/S0002-9947-2011-05308-0
- Published electronically: July 25, 2011
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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.References
- Walter Bergweiler and Alexandre Eremenko, Meromorphic functions with linearly distributed values and Julia sets of rational functions, Proc. Amer. Math. Soc. 137 (2009), no. 7, 2329–2333. MR 2495266, DOI 10.1090/S0002-9939-08-09788-8
- A. È. Erëmenko and M. Yu. Lyubich, The dynamics of analytic transformations, Algebra i Analiz 1 (1989), no. 3, 1–70 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 3, 563–634. MR 1015124
- P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 47 (1919), 161–271 (French). MR 1504787
- P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 208–314 (French). MR 1504797
- Anatoly A. Goldberg and Iossif V. Ostrovskii, Value distribution of meromorphic functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008. Translated from the 1970 Russian original by Mikhail Ostrovskii; With an appendix by Alexandre Eremenko and James K. Langley. MR 2435270, DOI 10.1090/mmono/236
- W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
- François Ledrappier, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 1, 37–40 (French, with English summary). MR 756305
- John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
- Rolf Nevanlinna, Analytic functions, Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig. MR 0279280
- J. F. Ritt, Periodic functions with a multiplication theorem, Trans. Amer. Math. Soc. 23 (1922), no. 1, 16–25. MR 1501186, DOI 10.1090/S0002-9947-1922-1501186-3
- Georges Valiron, Fonctions analytiques, Presses Universitaires de France, Paris, 1954 (French). MR 0061658
Bibliographic Information
- Alexandre Eremenko
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 63860
- 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
- 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. - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 6453-6463
- MSC (2010): Primary 37F10, 30D05
- DOI: https://doi.org/10.1090/S0002-9947-2011-05308-0
- MathSciNet review: 2833563