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Rational maps with real multipliers

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Eremenko, Alexandre and van Strien, Sebastian. (2011) Rational maps with real multipliers. Transactions of the American Mathematical Society, Vol.363 (No.12). pp. 6453-6463. ISSN 0002-9947

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Official URL: http://dx.doi.org/10.1090/S0002-9947-2011-05308-0

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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Mappings (Mathematics), Julia sets
Journal or Publication Title:	Transactions of the American Mathematical Society
Publisher:	American Mathematical Society
ISSN:	0002-9947
Date:	December 2011
Volume:	Vol.363
Number:	No.12
Page Range:	pp. 6453-6463
Identification Number:	10.1090/S0002-9947-2011-05308-0
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	National Science Foundation (U.S.) (NSF), Royal Society (Great Britain), Leverhulme Trust (LT)
Grant number:	DMS-0555279 (NSF)
References:	[1] W. Bergweiler and A. Eremenko, Meromorphic functions with linearly distributed values and Julia sets of rational functions, Proc. AMS, 137 (2009) 2329–2333. MR2495266 (2010d:30036) [2] A. Eremenko and M. Lyubich, Dynamics of analytic transformations, Leningrad Math. J., 1 (1990) 563–634. MR1015124 (91b:58109) [3] P. Fatou, Sur les ´equations fonctionnelles. Premiere m´emoire, Bull. Soc. Math. France, 47 (1919) 161–271. MR1504787 [4] P. Fatou, Sur les ´equations fonctionnelles. Troisi`eme m´emoire, Bull. Soc. Math. France, 48 (1920) 208–314. MR1504797 [5] A. Goldberg and I. Ostrovskii, Value distribution of meromorphic functions, AMS, Providence, RI, 2008. Troisieme memoire, Bull. Soc. Math. France, 48 (1920) 208–314. MR2435270 (2009f:30067) [6] W. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964. MR0164038 (29:1337) [7] F. Leddrapier, Quelques propri´et´es ergodiques des applications rationnelles, C. R. Acad. Sci., 299 (1984) 37–40. MR756305 (86c:58091) [8] J. Milnor, Dynamics in One Variable, Princeton Univ. Press, Princeton, NJ, 2006. MR2193309 (2006g:37070) [9] R. Nevanlinna, Analytic functions, Springer, NY, 1970. MR0279280 (43:5003) [10] J. F. Ritt, Periodic functions with a multiplication theorem, Trans. Amer. Math. Soc., 23 (1922) 16–25. MR1501186 [11] G. Valiron, Fonctions analytiques, Presses universitaires de France, Paris, 1954. MR0061658 (15:861a)
URI:	http://wrap.warwick.ac.uk/id/eprint/40411