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Iteration of a class of hyperbolic meromorphic functions

Author(s): P. J. Rippon; G. M. Stallard
Journal: Proc. Amer. Math. Soc. 127 (1999), 3251-3258.
MSC (1991): Primary 30D05
Posted: April 27, 1999
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Abstract: We look at the class which contains those transcendental meromorphic functions for which the finite singularities of $f^{-n}$ are in a bounded set and prove that, if belongs to , then there are no components of the set of normality in which $f^{mn}(z)\to\infty$ as $m\to\infty$ . We then consider the class $\widehat B$ which contains those functions in for which the forward orbits of the singularities of $f^{-1}$ stay away from the Julia set and show (a) that there is a bounded set containing the finite singularities of all the functions $f^{-n}$ and (b) that, for points in the Julia set of , the derivatives have exponential-type growth. This justifies the assertion that $\widehat B$ is a class of hyperbolic functions.

References:

1.: I. N. Baker, J. Kotus and L. Yinian, Iterates of meromorphic functions II. J. London Math. Soc. (2) 42 (1990), 267-278. MR 92m:58114
2.: A. F. Beardon, Iteration of rational functions, Springer, 1991. MR 92j:30026
3.: W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151-188. MR 94c:30033
4.: L. Carleson and T. W. Gamelin, Complex dynamics, Springer, 1993. MR 94h:30033
5.: P. L. Duren, Univalent functions, Springer, 1983. MR 85j:30034
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7.: M. E. Herring, An extension of the Julia-Fatou theory of iteration, Ph.D. thesis, University of London, 1994.
8.: G. M. Stallard, The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions, To appear in Math. Proc. Camb. Phil. Soc.

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Additional Information:

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

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

DOI: 10.1090/S0002-9939-99-04942-4
PII: S 0002-9939(99)04942-4
Received by editor(s): September 30, 1997
Received by editor(s) in revised form: January 26, 1998
Posted: April 27, 1999
Dedicated: Dedicated to Professor Noel Baker on the occasion of his retirement
Communicated by: Mary Rees
Copyright of article: Copyright 1999, American Mathematical Society


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