Transactions of the American Mathematical Society

Author(s): Walter Bergweiler; Norbert Terglane
Journal: Trans. Amer. Math. Soc. 348 (1996), 1-12.
MSC (1991): Primary 30D05, 58F23
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: It is proved that if a transcendental meromorphic function $f$

has a multiply-connected wandering domain, then $f$

has a fixpoint $z_0$

such that

. Entire functions with a multiply-connected wandering domain have infinitely many such fixpoints. These results are used to show that solutions of certain differential equations do not have wandering domains at all.

1: L. V. Ahlfors, Conformal invariants, McGraw-Hill, New York, 1973. MR 50:10211
2: I. N. Baker, An entire function which has wandering domains, J. Australian Math. Soc. Ser. A 22 (1976), 173-176. MR 54:7777
3: I. N. Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc. (3) 49 (1984), 563-576. MR 86d:58066
4: I. N. Baker, Some entire functions with multiply-connected wandering domains, Ergod. Th. and Dynam. Sys. 5 (1985), 163-169. MR 86i:30031
5: I. N. Baker, Iteration of entire functions: an introductory survey, in Lectures on complex analysis, World Scientific, Singapore, New Jersey, London, Hong Kong, 1987, 1-17. MR 90d:30072
6: I. N. Baker, Wandering domains for maps of the punctured plane, Ann. Acad. Sci. Fen. Ser. A I 12 (1987), 191-198. MR 89g:30046
7: I. N. Baker, J. Kotus, and Y. Lü, Iterates of meromorphic functions I, Ergod. Th. and Dynam. Sys. 11 (1991), 241-248. MR 92m:58113
8: I. N. Baker, J. Kotus, and Y. Lü, Iterates of meromorphic functions II: Examples of wandering domains, J. London Math. Soc. (2) 42 (1990), 267-278. MR 92m:58114
9: I. N. Baker, J. Kotus, and Y. Lü, Iterates of meromorphic functions III: Preperiodic domains, Ergod. Th. and Dynam. Sys. 11 (1991), 603-618. MR 92m:58115
10: I. N. Baker, J. Kotus, and Y. Lü, Iterates of meromorphic functions IV: Critically finite functions, Results Math. 22 (1992), 651-656. MR 94c:58166
11: S. B. Bank and R. P. Kaufman, On the growth of meromorphic solutions of the differential equation , Acta Math. 144 (1980), 223-248. MR 81k:30028
12: A. F. Beardon, Iteration of rational functions, Springer, New York, Berlin, Heidelberg, 1991. MR 92j:30026
13: W. Bergweiler, Newton's method and a class of meromorphic functions without wandering domains, Ergod. Th. and Dynam. Sys. 13 (1993), 231-247. CMP 93:17
14: W. Bergweiler, M. Haruta, H. Kriete, H.-G. Meier and N. Terglane, On the limit functions of iterates in wandering domains, Ann. Acad. Sci. Fenn. 18 (1993), 369-375. MR 94j:30023
15: W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151-188. MR 94c:30033
16: P. Bhattacharyya, Boundaries of domains of attraction, Publ. Math. Debrecen 18 (1971), 95-98. MR 46:5626
17: R. L. Devaney and L. Keen, Dynamics of meromorphic maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. (4) 22 (1989), 55-81. MR 90e:58071
18: A. E. Eremenko and M. Yu. Lyubich, Examples of entire functions with pathological dynamics, J. London Math. Soc. (2) 36 (1987), 458-468. MR 89e:30047
19: A. E. Eremenko and M. Yu. Lyubich, The dynamics of analytic transforms, Leningrad Math. J. 1 (1990), 563-634. MR 91b:58109
20: A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier 42 (1992), 989-1020. MR 93k:30034
21: P. Fatou, Sur les équations fonctionelles, Bull. Soc. Math. France 47 (1919), 161-271; 48 (1920), 33-94, 208-314.
22: P. Fatou, Sur l'itération des fonctions transcendantes entières, Acta Math. 47 (1926), 337-360.
23: L. R. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergod. Th. and Dynam. Sys. 6 (1986), 183-192. MR 88b:58126
24: G. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), 88-104. MR 88m:30076
25: W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964. MR 29:1337
26: M. Herman, Exemples de fractions rationelles ayant une orbite dense sur la sphère de Riemann, Bull. Soc. Math. France 112 (1984), 93-142. MR 86d:58055
27: G. Jank and L. Volkmann, Ein elementarer Beweis des Satzes von Malmquist-Yosida-Steinmetz, Complex Variables Theory Appl. 1 (1983), 181-194. MR 85f:34004
28: G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser, Basel, Boston, Stuttgart, 1985. MR 87h:30066
29: G. Julia, Sur l'itération des fonctions rationelles, J. Math. Pures Appl. (7) 4 (1918), 47-245.
30: J. Milnor, Dynamics in one complex variable: introductory lectures, Stony Brook Institute for Mathematical Sciences, Preprint 1990/5.
31: R. Nevanlinna, Analytic functions, Springer, Berlin, Heidelberg, New York, 1970. MR 43:5003
32: F. Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps, Dynamical systems and ergodic theory, Banach Center Publ. 23 (1989), 229-235. MR 92e:58180
33: M. Shishikura, On the quasi-conformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), 1-29. MR 88i:58099
34: M. Shishikura, The connectivity of the Julia set and fixed points, Preprint IHES-M-90-37 (1990).
35: G. M. Stallard, A class of meromorphic functions with no wandering domains, Ann. Acad. Sci. Fenn. 16 (1991), 211-226. MR 93e:30059
36: N. Steinmetz, Eigenschaften eindeutiger Lösungen gewöhnlicher Differentialgleichungen im Komplexen, Dissertation, Karlsruhe 1978.
37: N. Steinmetz, Rational iteration, Walter de Gruyter, Berlin, 1993. MR 94h:30035
38: D. Sullivan, Itération des fonctions analytiques complexes, C. R. Acad. Sci. Paris 294 (1982), 301-303. MR 83d:58061
39: D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Ann. Math. 122 (1985), 401-418. MR 87i:58103
40: J. E. Whittington, On the fixpoints of entire functions, Proc. London Math. Soc. (3) 17 (1967), 530-546. MR 35:5616

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 30D05, 58F23

Walter Bergweiler
Affiliation: Lehrstuhl II für Mathematik, RWTH Aachen, D-52056 Aachen, Germany
Address at time of publication: Fachbereich Mathematik, Sekr. MA 8--2, TU Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany
Email: bergweil@math.tu-berlin.de

Norbert Terglane
Affiliation: Lehrstuhl II für Mathematik, RWTH Aachen, D-52056 Aachen, Germany
Email: terglan@math2.rwth-aachen.de

DOI: 10.1090/S0002-9947-96-01511-5
PII: S 0002-9947(96)01511-5
Received by editor(s): August 17, 1993
Copyright of article: Copyright 1996, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS