manuscripta math. 74, 299 - 319 (1992) manuscripta mathematica 9 Springer-Verlag 1992 Snapback Repellers and Semiconjugacy for Iterated Entire Mappings * Global aspects of an old Theorem of Poincar4 F. Rothe Abstract: A theorem of Poincar4 guarantees existence of the local conjugacy of an entire analytic mapping with an hyperbolically unstable fixed point to the linearized mapping. Since the local conjugacy can be extended to a global conjugacy, it is a valuable tool for the global study of dynamics. Especially we focus on snapback repellers which are defined as entire orbits which tend to an unstable fixed point in the past and snap back to the same fixed point. Snapback repellers correspond to the zeros of the semiconjugacy. It turns out that in general there exist infinitely many snapback points and for each one of them there exist infinitely many snapback repellers. The exceptional classes of functions with a different behavior are characterized. The proof exploits the Theorem of Picard about the range of values that an analytic function assumes near an essential singularity. Furtheron, we relate the multiplicity of the zeros of the semiconjugacy to the occurence of critical points in the corresponding snapback repeller. For quadratic mappings and their iterates, the zeros of the semiconjugacy have at most multiplicity two. Key words: Iterated mappings, unstable fixed points, periodic orbits, semicon- jugacy equation, snapback repellers, critical points, Theorem of Picard, lacunary value. AMS(MOS) subject classifications: 30D05, 58F23 * This work has been supported in part by the EC-project 89400552/JU1: "Evolutionary systems" 299