CONNECTIVITY OF JULIA SETS OF TRANSCENDENTAL MEROMORPHIC MAPS Nuria Fagella*, Xavier Jarque*, Jordi Taixes* *Departament de Matematica aplicada i Analisi Universitat de Barcelona Gran Via 585 08005 Barcelona Spain e-mail: fagella@maia.ub.es e-mail: xavier.jarque@ub.edu e-mail: taixes@maia.ub.es ABSTRACT It is known that the Julia set of the Newton's method of a non-constant polynomial is connected (\cite{shishikura}). 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, namely, we show the existence of such fixed points provided that immediate attractive basins or preperiodic components be multiply connected.