Snapback repellers and semiconjugacy for iterated entire mappings. Global aspects of an old theorem of Poincaré
A theorem of Poincaré guarantees the existence of the local conjugacy of an entire analytic mapping with a hyperbolically unstable fixed point to the horizontal mapping. Since the local conjugacy can be extended to a global conjugacy, it is a valuable tool for the global study of dynamics. The author especially focuses on snapback repellers which are defined as entire orbits which tend to an unstable fixed point in the past and snapback to the same fixed point. Snapback repellers correspond to the zeros of the semiconjugacy. In general there exist infinitely many snapback points and for each of these there exist infinite 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, the author relates the multiplicity of the zeros on the semiconjugacy to the occurrence of critical points in the corresponding snapback repeller. For quadratic mappings and their iterates, the zeros of the semiconjugacy have at most a multiplicity of two.
scientific article; zbMATH DE number 60074