Selections and their Absolutely Continuous Invariant Measures

Let $I=[0,1]$ and consider disjoint closed regions $G_1,\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.},G_n$ in $% I\times I$andsubintervals$I_{1},…,I_{n},$suchthat$G_{i}$ projects onto $I_{i\mathrm{.}}$ We define the lower and upper maps ${\tau }_1,$ ${\tau }_2$ by the lower and upper boundaries of $G_i,i=1,\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.},n,$ respectively. We assume ${\tau }_1$, ${\tau }_2$ to be piecewise monotonic and preserving continuous invariant measures ${\mu }_1$ and ${\mu }_2$, respectively. Let $$ and $F^{(2)}$ be the distribution functions of ${\mu }_1$ and ${\mu }_2\mathrm{.}$ The main results shows that for any convex combination $F$ of $$ 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.