Absolutely continuous SRB measures for random Lasota-Yorke maps

A. Lasota and J. A. Yorke proved that a piecewise expanding interval map admits finitely many ergodic absolutely continuous invariant probability measures. We generalize this to the random composition of such maps under conditions which are natural and less restrictive than those previously studied by Morita and Pelikan. For instance our conditions are satisfied in the case of arbitrary random beta-transformations, i.e., x bar right arrow beta x mod 1 on [0, 1] where beta is chosen according to any stationary stochastic process (in particular, not necessarily i.i.d.) with values in ]1, infinity[.