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Singularities and Limit Functions in Iteration of Meromorphic Functions
Author(s) -
Zheng Jian-Hua
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003800
Subject(s) - meromorphic function , mathematics , gravitational singularity , limit (mathematics) , julia set , entire function , pure mathematics , domain (mathematical analysis) , function (biology) , transcendental number , limit set , mathematical analysis , evolutionary biology , biology
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 { f n | U } lie in the derived set of P(f) and that if f n p | V → q ( n → +∞) for a Fatou component V , then either q is in the derived set of S p ( f ) or f p ( q ) = q . As applications of main theorems, some sufficient conditions of the non‐existence of wandering domains and Baker domains are given.