On connectivity of Julia sets of transcendental meromorphic maps and weakly repelling fixed points I

摘要

It is known that the Julia set of the Newton method of a non-constant polynomial is connected (Mitsuhiro Shishikura, Preprint, 1990, M/90/37, Inst. Hautes Etudes Sci.). This is, in fact, a consequence of a much more general result that establishes the relationship between simple connectivity of Fatou components of rational maps and fixed points which are repelling or parabolic with multiplier 1. In this paper we study Fatou components of transcendental meromorphic functions; that is, we show the existence of such fixed points, provided that immediate attractive basins or preperiodic components are multiply connected.