  Recently for a class of critically intermittent random systems a phase
transition was found for the finiteness of the absolutely continuous invariant
measure. The systems for which this result holds are characterized by the
interplay between a superexponentially attracting fixed point and an
exponentially repelling fixed point. In this article we consider a closely
related family of random systems with instead exponentially fast attraction to
and polynomially fast repulsion from two fixed points, and show that such a
phase transition still exists. The method of the proof however is different and
relies on the construction of a suitable invariant set for the transfer
operator.
