  Motivated by the study of the time evolution of random dynamical systems
arising in a vast variety of domains --- ranging from physics to ecology ---,
we establish conditions for the occurrence of a non-trivial asymptotic
behaviour for these systems in the absence of an ellipticity condition. More
precisely, we classify these systems according to their type and --- in the
recurrent case --- provide with sharp conditions quantifying the nature of
recurrence by establishing which moments of passage times exist and which do
not exist. The problem is tackled by mapping the random dynamical systems into
Markov chains on $\mathbb{R}$ with heavy-tailed innovation and then using
powerful methods stemming from Lyapunov functions to map the resulting Markov
chains into positive semi-martingales.
