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On questions of Fatou and Eremenko


Authors: P. J. Rippon and G. M. Stallard
Journal: Proc. Amer. Math. Soc. 133 (2005), 1119-1126
MSC (2000): Primary 37F10; Secondary 37F45
DOI: https://doi.org/10.1090/S0002-9939-04-07805-0
Published electronically: October 18, 2004
MathSciNet review: 2117213
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $f$ be a transcendental entire function and let $I(f)$ be the set of points whose iterates under $f$ tend to infinity. We show that $I(f)$ has at least one unbounded component. In the case that $f$ has a Baker wandering domain, we show that $I(f)$is a connected unbounded set.


References [Enhancements On Off] (What's this?)

  • 1. I.N. Baker. An entire function which has wandering domains. J. Austral. Math. Soc. Ser. A 22 (1976), 173-176. MR 0419759 (54:7777)
  • 2. I.N. Baker. Wandering domains in the iteration of entire functions. Proc. London Math. Soc. (3) 49 (1984), 563-576. MR 0759304 (86d:58066)
  • 3. W. Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc., 29 (1993), 151-188. MR 1216719 (94c:30033)
  • 4. W. Bergweiler. Invariant domains and singularities. Math. Proc. Camb. Phil. Soc., 117 (1995), 525-532. MR 1317494 (96b:30055)
  • 5. W. Bergweiler and A. Hinkkanen. On semiconjugation of entire functions. Math. Proc. Camb. Phil. Soc., 126 (1999), 565-574. MR 1684251 (2000c:37057)
  • 6. R.L. Devaney and F. Tangerman. Dynamics of entire functions near the essential singularity. Ergod. Th. and Dynam. Sys., 6 (1986), 489-503. MR 0873428 (88e:58057)
  • 7. A.E. Eremenko. On the iteration of entire functions. Dynamical systems and ergodic theory, Banach Center Publ. 23 (Polish Scientific Publishers, Warsaw, 1989) 339-345. MR 1102727 (92c:30027)
  • 8. P. Fatou. Sur l'itération des fonctions transcendantes entières. Acta Math., 47 (1926), 337-370.
  • 9. W.K Hayman. Meromorphic functions. Clarendon Press, Oxford, 1964. MR 0164038 (29:1337)
  • 10. M. Kisaka, On the connectivity of Julia sets of transcendental entire functions. Ergodic Theory Dynam. Systems, 18 (1998), 189-205. MR 1609471 (99a:30033)
  • 11. M.H.A. Newman. Elements of the topology of plane sets of points, Cambridge University Press, 1961. MR 0132534 (24A:2374)
  • 12. D. Schleicher and J. Zimmer. Escaping points of exponential maps. J. London Math. Soc. (2), 67 (2003), 380-400. MR 1956142 (2003k:37067)
  • 13. P.J. Rippon and G.M. Stallard. On sets where iterates of a meromorphic function zip towards infinity. Bull. London Math. Soc., 32 (2000), 528-536. MR 1767705 (2001g:30019)

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Additional Information

P. J. Rippon
Affiliation: Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
Email: p.j.rippon@open.ac.uk

G. M. Stallard
Affiliation: Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
Email: g.m.stallard@open.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-04-07805-0
Received by editor(s): April 4, 2003
Received by editor(s) in revised form: November 28, 2003
Published electronically: October 18, 2004
Dedicated: This paper is dedicated to the memory of Professor Noel Baker
Communicated by: Michael Handel
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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