In this talk we will describe an intermittent behaviour for random dynamical systems on an interval caused
by the existence of a superattracting fixed point. The systems we consider are i.i.d. compositions of a finite
number of maps of two types: 'bad' maps which share a superattracting fixed point and 'good' maps that
map the superattacting fixed point onto a common repelling fixed point.
We show that for a fairly big class of good maps there exists a sigma finite absolutely continuous stationary
measure, which is ergodic. Further, we obtain necessary and sufficient condition for the stationary measure
to be finite in terms of the probabilities of choosing the bad maps and the order of criticality at the
superattracting fixed point.
This is a joint work in progress with A.-J. Homburg, Ch. Kalle, E.Verbitskiy and B. Zeegers.