### Uniformly perfect Julia sets of meromorphic functions

Sheng Wang, Liang-Wen Liao
2005 Bulletin of the Australian Mathematical Society
Julia sets of meromorphic functions are uniformly perfect under some suitable conditions. So are Julia sets of the skew product associated with finitely generated semigroup of rational functions. Pommerenke and Beardon [5, 12, 13] introduced and studied the uniformly perfect sets. Following Pommerenke, many authors have researched the topic, and many papers on the uniformly perfect sets have appeared in the literature. It was proved that the Julia sets of rational functions are uniformly
more » ... , see [8, 9, 11] , and also  . However, unlike the case of rational functions, the Julia sets of a transcendental meromorphic function may not be uniformly perfect, see the example in Section 1. But, it is interesting to discuss when the Julia sets of transcendental meromorphic functions will be uniformly perfect. Zheng [17, 18, 19 ] studied uniformly perfect boundaries of stable domains in the iteration of meromorphic functions. In this paper, we shall study the Julia sets of functions from the class PM, which a is more general class of functions than meromorphic functions. We shall also prove that the Julia sets of the skew product associated with a finitely generated semigroup of rational functions, each generator having degree not less than 2, are uniformly perfect. UNIFORMLY PERFECT JULIA SETS ON CLASS PM Following Baker, Dominguez and Herring  , let E be a compact totally disconnected set in the Riemann sphere C = CU {oo}, and f(z) be a function meromorphic in E c = C\E and such that C(f, E c , z 0 ) = C for z 0 € E, where the cluster C{f, E c , z 0 ) is defined by n-•oo J Denote by PM the set of all functions satisfying the condition that there is a compact totally disconnected set E = E(f) such that f(z) is meromorphic in E c and C(f, E c , z 0 ) = C for all 2 0 € E.