Iteration of a class of hyperbolic meromorphic functions

Authors:
  • PRP. Rippon
    P. Rippon
  • GSG. Stallard
    G. Stallard
Publisher website
Google Scholar
Abstract
We look at the class B n B_n which contains those transcendental meromorphic functions f f for which the finite singularities of f − n f^{-n} are in a bounded set and prove that, if f f belongs to B n B_n , then there are no components of the set of normality in which f m n ( z ) → ∞ f^{mn}(z)\to \infty as m → ∞ m\to \infty . We then consider the class B ^ \widehat B which contains those functions f f in B 1 B_1 for which the forward orbits of the singularities of f − 1 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 f^{-n} and (b) that, for points in the Julia set of f f , the derivatives ( f n ) ′ (f^n)’ have exponential-type growth. This justifies the assertion that B ^ \widehat B is a class of hyperbolic functions.

This publication has 5 references indexed in Scilit: