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Journal of the London Mathematical Society 2003 67(1):195-207; doi:10.1112/S0024610702003800
© 2003 by London Mathematical Society
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© The London Mathematical Society

Singularities and Limit Functions in Iteration of Meromorphic Functions

Jian-Hua Zheng

Department of Mathematical Sciences, Tsinghua University Beijing 100084, China, jzheng{at}math.tsinghua.edu.cn

Received 28 March 2001. Revision received 21 March 2002.

Let f(z) be a transcendental meromorphic function. The paper investigates, using the hyperbolic metric, the relation between the forward orbit P(f) of singularities of f–1 and limit functions of iterations of f in its Fatou components. It is mainly proved, among other things, that for a wandering domain U, all the limit functions of {fn|U} lie in the derived set of P(f) and that if fnp|V-> q(n-> +{infty}) for a Fatou component V, then either q is in the derived set of Sp (f) or fp(q) = q. As applications of main theorems, some sufficient conditions of the non-existence of wandering domains and Baker domains are given.


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