Unbounded Not Diverging Trajectories in Maps with a Vanishing Denominator

Maps with a denominator which vanishes in a subset of the phase space may generate unbounded trajectories which are not divergent, i.e. trajectories involving arbitrarily large values of the dynamic variables but which are not attracted to infinity. In this paper we propose some simple one-dimensional and two-dimensional recurrences which generate unbounded chaotic sequences, and through these examples we try to explain the basic mechanisms and bifurcations leading to the creation of unbounded sets of attraction.