On the number of invariant measures for random expanding maps in higher dimensions

In [22], Jabłoński proved that a piecewise expanding \begin{document}$ C^{2} $\end{document} multidimensional map admits an absolutely continuous invariant probability measure (ACIP). In rid="b6">6], Boyarsky and Lou extended this result to the case of i.i.d. compositions above maps, with on average condition. We generalize these results (quenched) setting random where randomness is governed by ergodic, invertible preserving transformation. prove skew product associated dynamical system finite number ergodic ACIPs. Furthermore, we provide two different upper bounds mutually singular ACIPs, motivated works Buzzi rid="b9">9] in one dimension Góra, Proppe rid="b19">19] higher dimensions.