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 $τ_{1},$ $τ_{2}$ by the lower and upper boundaries of $G_{i},i=1,....,n,$ respectively. We assume $τ_{1}$, $τ_{2}$ to be piecewise monotonic and preserving continuous invariant measures $μ_{1}$ and $μ_{2}$, respectively. Let $% F^{(1)}$ and $F^{(2)}$ be the distribution functions of $μ_{1}$ and $μ_{2}.$ The main results shows that for any convex combination $F$ of $% F^{(1)} $ and $F^{(2)}$ we can find a map $η$ with values between the graphs of $τ_{1}$ and $τ_{2}$ (that is, a selection) such that $F$ is the $η$-invariant distribution function. Examples are presented. We also study the relationship of the dynamics of multi-valued maps to random maps.