This is a joint work in progress with A.-J. Homburg, Ch. Kalle, E. Verbitskiy and B. Zeegers.
Abstract: 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 superattracting 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 critically at the superattracting fixed point.