Unbounded sets of attraction

In this paper we show that unbounded chaotic trajectories are easily observed in the iteration of maps which are not defined everywhere, due to the presence of a denominator which vanishes in a zero-measure set. Through simple examples, obtained by the iteration of one-dimensional and two-dimensional maps with denominator, the basic mechanisms which are at the basis of the existence of unbounded chaotic trajectories are explained. Moreover, new kinds of contact bifurcations, which mark the transition from bounded to unbounded sets of attraction, are studied both through the examples and by general theoretical methods. Some of the maps studied in this paper have been obtained by a method based on the Schroder functional equation, which allows one to write closed analytical expressions of the unbounded chaotic trajectories, in terms of elementary functions.