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| Let $I=[0,1]$ and let $P$ be a partition of $I$ into a finite number of
intervals. Let $\tau _{1},$ $\tau _{2}; I\rightarrow I$ be two piecewise
expanding maps on $P.$ Let $G$ $\subset I\times I$ be the region between the
boundaries of the graphs of $\tau _{1}$ and $\tau _{2}.$ Any map $\tau
:I\rightarrow I$ that takes values in $G$ is called a selection of the
multivalued map defined by $G.$ There are many results devoted to the study
of the existence of selections with specified topological properties.
However, there are no results concerning the existence of selection with
measure-theoretic properties. In this paper we prove the existence of
selections which have absolutely continuous invariant measures (acim). By
our assumptions we know that $\tau _{1}$ and $\tau _{2}$ possess acims
preserving the distribution functions $F^{(1)}$ and $F^{(2)}.$
The main result 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 our multivalued maps to random maps. |
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