Unbounded sets of attraction

In this paper we show that unbounded chaotic trajectories are easily observ ed in the iteration of maps which are not defined everywhere, due to the pr esence of a denominator which vanishes in a zero-measure set. Through simpl e examples, obtained by the iteration of one-dimensional and two-dimensiona l maps with denominator, the basic mechanisms which are at the basis of the existence of unbounded chaotic trajectories are explained. Moreover, new k inds of contact bifurcations, which mark the transition from bounded to unb ounded sets of attraction, are studied both through the examples and by gen eral theoretical methods. Some of the maps studied in this paper have been obtained by a method based on the Schroder functional equation, which allow s one to write closed analytical expressions of the unbounded chaotic traje ctories, in terms of elementary functions.