  In \cite{J}, Jab\l o\'{n}ski proved that a piecewise expanding $C^{2}$
multidimensional Jab\l o\'{n}ski map admits an absolutely continuous invariant
probability measure (ACIP). In \cite{BL}, Boyarsky and Lou extended this result
to the case of i.i.d. compositions of the above maps, with an on average
expanding condition. We generalize these results to the (quenched) setting of
random Jab\l o\'{n}ski maps, where the randomness is governed by an ergodic,
invertible and measure preserving transformation. We prove that the skew
product associated to this random dynamical system admits a finite number of
ergodic ACIPs. Furthermore, we provide two different upper bounds on the number
of mutually singular ergodic ACIP's, motivated by the works of Buzzi \cite{B}
in one dimension and Gora, Boyarsky and Proppe \cite{GBP} in higher dimensions.
