## Unbounded domains of normality

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- by J. M. Anderson and A. Hinkkanen PDF
- Proc. Amer. Math. Soc.
**126**(1998), 3243-3252 Request permission

## Abstract:

Let $f$ be a transcendental entire function of order less than 1/2. We introduce the method of “self-sustaining spread” to study the components of the set of normality of such a function. We give a new proof of the fact that any preperiodic or periodic component of the set of normality of $f$ is bounded. We obtain the same conclusion for a wandering domain if the growth rate of $f$ is never too small.## References

- Irvine Noel Baker,
*Zusammensetzungen ganzer Funktionen*, Math. Z.**69**(1958), 121–163 (German). MR**97532**, DOI 10.1007/BF01187396 - I. N. Baker,
*Fixpoints and iterates of entire functions*, Math. Z.**71**(1959), 146–153. MR**107015**, DOI 10.1007/BF01181396 - I. N. Baker,
*The iteration of polynomials and transcendental entire functions*, J. Austral. Math. Soc. Ser. A**30**(1980/81), no. 4, 483–495. MR**621564**, DOI 10.1017/S1446788700017961 - I. N. Baker,
*Wandering domains in the iteration of entire functions*, Proc. London Math. Soc. (3)**49**(1984), no. 3, 563–576. MR**759304**, DOI 10.1112/plms/s3-49.3.563 - P. D. Barry,
*On a theorem of Besicovitch*, Quart. J. Math. Oxford Ser. (2)**14**(1963), 293–302. MR**156993**, DOI 10.1093/qmath/14.1.293 - Alan F. Beardon,
*Iteration of rational functions*, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR**1128089**, DOI 10.1007/978-1-4612-4422-6 - W. Bergweiler,
*The iteration of meromorphic functions*, Bull. Amer. Math. Soc.**29**(1993), 151–188. - P. Fatou,
*Sur les équations fonctionnelles*, Bull. Soc. Math. France**47**(1919), 161–271. - —,
*Sur les équations fonctionnelles*, Bull. Soc. Math. France**48**(1920), 33–94, 208–314. - —,
*Sur l’itération des fonctions transcendantes entières*, Acta Math.**47**(1926), 337–370. - G. Julia,
*Mémoire sur l’itération des fonctions rationnelles*, J. Math. Pures Appl.**8**(1918), 47–245. - G. Stallard,
*Some problems in the iteration of meromorphic functions*, Ph.D. thesis, Imperial College, London, 1991. - Gwyneth M. Stallard,
*The iteration of entire functions of small growth*, Math. Proc. Cambridge Philos. Soc.**114**(1993), no. 1, 43–55. MR**1219913**, DOI 10.1017/S0305004100071395

## Additional Information

**J. M. Anderson**- Affiliation: Department of Mathematics, University College, London WC1E 6BT, United Kingdom
**A. Hinkkanen**- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- MR Author ID: 86135
- Received by editor(s): July 8, 1996
- Received by editor(s) in revised form: March 10, 1997
- Additional Notes: This research was completed while the first author was visiting the University of Illinois at Urbana-Champaign. He wishes to thank the Department of Mathematics for its kind hospitality.

The research of the second author was partially supported by the Alfred P. Sloan Foundation, by the U.S. National Science Foundation grant DMS 94-00999, and by the U.S. National Security Agency grant MDA904-95-H-1014. - Communicated by: Albert Baernstein II
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**126**(1998), 3243-3252 - MSC (1991): Primary 30D05, 58F23
- DOI: https://doi.org/10.1090/S0002-9939-98-04370-6
- MathSciNet review: 1452790