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

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