Iteration of a class of hyperbolic meromorphic functions
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- by P. J. Rippon and G. M. Stallard PDF
- Proc. Amer. Math. Soc. 127 (1999), 3251-3258 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
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Additional 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