Selections and their absolutely continuous invariant measures

Let I=[0,1]I=[0,1] and let P be a partition of I   into a finite number of intervals. Let τ1τ1, τ2τ2; I→II→I be two piecewise expanding maps on P  . Let G⊂I×IG⊂I×I be the region between the boundaries of the graphs of τ1τ1 and τ2τ2. Any map τ:I→Iτ:I→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 τ1τ1 and τ2τ2 possess acims preserving the distribution functions F(1)F(1) and F(2)F(2). The main result shows that for any convex combination F   of F(1)F(1) and F(2)F(2) we can find a map η   with values between the graphs of τ1τ1 and τ2τ2 (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 our multivalued maps to random maps.