New phenomena related to the presence of focal points in two-dimensional maps - Bischi, Gian-Italo; Gardini, Laura; Mira, Christian (Annales Mathematicae Silesianae, 1999)
In this paper we consider two-dimensional maps, defined in the whole plane, with the property of mapping a whole curve δ into a single point Q. We relate such property to the fact that at least one inverse map exists with a denominator which can vanish, and assumes the form 0/0 in Q. This allows us to apply the properties of focal points and prefocal curves in order to explain the dynamic phenomena observed. By an example we show that the iteration of such maps may generate discrete dynamical systems with peculiar attracting sets, characterized by the presence of "knots", where infinitely many phase curves shrink into a single point.