Inverse Poincare Halfmaps

A family of critical curves t h a t lie in the r ange of a class of Po inca re h a l f m a p s induced by a saddle focus in three-space is inves t iga ted analyt ical ly . It compr i s e s all " f a r f inal po in t s " of the images of invar iant curves tha t a re subjec t to a s e p a r a t i n g m e c h a n i s m . T h e n u m b e r of b ranches occurr ing on a speci f ic critical cu rve is direct ly re la ted to the type of sepa ra t ing m e c h a n i s m present in the h a l f m a p . M o r e o v e r , universal p rope r t i e s of the sadd le focus d y n a m i c s are demons t ra ted for the first t ime.