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Iteration of a class of hyperbolic
meromorphic functions


Authors: P. J. Rippon and G. M. Stallard
Journal: Proc. Amer. Math. Soc. 127 (1999), 3251-3258
MSC (1991): Primary 30D05
DOI: https://doi.org/10.1090/S0002-9939-99-04942-4
Published electronically: April 27, 1999
MathSciNet review: 1610785
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Abstract | References | Similar Articles | Additional Information

Abstract: We look at the class $B_n$ which contains those transcendental meromorphic functions $f$ for which the finite singularities of $f^{-n}$ are in a bounded set and prove that, if $f$ belongs to $B_n$, 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 $f$ in $B_1$ 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 $f$, the derivatives $(f^n)'$ have exponential-type growth. This justifies the assertion that $\widehat B$ is a class of hyperbolic functions.


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

  • 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
  • 6. A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier, Grenoble 42 (1992), 989-1020. MR 93k:30034
  • 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: https://doi.org/10.1090/S0002-9939-99-04942-4
Received by editor(s): September 30, 1997
Received by editor(s) in revised form: January 26, 1998
Published electronically: April 27, 1999
Dedicated: Dedicated to Professor Noel Baker on the occasion of his retirement
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society

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