Absolutely continuous invariant measures for position dependent random maps of piecewise real-analytic maps on the plane

Let {τ1, τ2, · · · , τK} be a collection of piecewise real analytic maps on a real analytic partition P of a bounded region S ⊂ R2 . Let {p1, p2, · · · , pK} be a collection of position dependent probabilities which are piecewise C1 on P. We consider position dependent random maps T = {τ1, τ2, · · · , τK; p1, p2, · · · , pK}. . In this paper, we prove the existence of absolutely continuous invariant measures of T.