Selections and their Absolutely Continuous Invariant Measures

Let $I=[0,1]$ and consider disjoint closed regions $G_{1},....,G_{n}$ in $% I\times I$ and subintervals $I_{1},......,I_{n},$ such that $G_{i}$ projects onto $I_{i.}$ We define the lower and upper maps $\tau_{1},$ $\tau_{2}$ by the lower and upper boundaries of $G_{i},i=1,....,n,$ respectively. We assume $\tau_{1}$, $\tau_{2}$ to be piecewise monotonic and preserving continuous invariant measures $\mu_{1}$ and $\mu_{2}$, respectively. Let $% F^{(1)}$ and $F^{(2)}$ be the distribution functions of $\mu_{1}$ and $\mu_{2}.$ The main results shows that for any convex combination $F$ of $% F^{(1)} $ and $F^{(2)}$ we can find a map $\eta $ with values between the graphs of $\tau_{1}$ and $\tau_{2}$ (that is, a selection) such that $F$ is the $\eta $-invariant distribution function. Examples are presented. We also study the relationship of the dynamics of multi-valued maps to random maps.