Iteration of a class of hyperbolic meromorphic functions
HTML articles powered by AMS MathViewer
- by P. J. Rippon and G. M. Stallard
- Proc. Amer. Math. Soc. 127 (1999), 3251-3258
- DOI: https://doi.org/10.1090/S0002-9939-99-04942-4
- Published electronically: April 27, 1999
- PDF | Request permission
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
- I. N. Baker, J. Kotus, and Yi Nian Lü, Iterates of meromorphic functions. II. Examples of wandering domains, J. London Math. Soc. (2) 42 (1990), no. 2, 267–278. MR 1083445, DOI 10.1112/jlms/s2-42.2.267
- 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
- Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188. MR 1216719, DOI 10.1090/S0273-0979-1993-00432-4
- Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
- Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
- A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020 (English, with English and French summaries). MR 1196102, DOI 10.5802/aif.1318
- M. E. Herring, An extension of the Julia-Fatou theory of iteration, Ph.D. thesis, University of London, 1994.
- G. M. Stallard, The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions, To appear in Math. Proc. Camb. Phil. Soc.
Bibliographic Information
- P. J. Rippon
- Affiliation: Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England
- MR Author ID: 190595
- 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
- MR Author ID: 292621
- Email: g.m.stallard@open.ac.uk
- Received by editor(s): September 30, 1997
- Received by editor(s) in revised form: January 26, 1998
- Published electronically: April 27, 1999
- Communicated by: Mary Rees
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 3251-3258
- MSC (1991): Primary 30D05
- DOI: https://doi.org/10.1090/S0002-9939-99-04942-4
- MathSciNet review: 1610785
Dedicated: Dedicated to Professor Noel Baker on the occasion of his retirement