Selections and their Absolutely Continuous Invariant Measures

Abstract

Let $I=[0,1]$ and consider disjoint closed regions $G_1,....,G_n$ in 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 and $F^{(2)}$ be the distribution functions of ${\mu }_1$ and ${\mu }_2.$ 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.