Rational maps with real multipliers

Eremenko, Alexandre; van Strien, Sebastian

doi:10.1090/S0002-9947-2011-05308-0

Rational maps with real multipliers
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by Alexandre Eremenko and Sebastian van Strien PDF

Trans. Amer. Math. Soc. 363 (2011), 6453-6463 Request permission

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