Selections and their Absolutely Continuous Invariant Measures

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Let I=[0,1] and consider disjoint closed regions G₁,....,Gₙ in % I× I and subintervals I₁,......,Iₙ, such that Gᵢ projects onto Iᵢ. We define the lower and upper maps τ₁, τ₂ by the lower and upper boundaries of Gᵢ,i=1,....,n, respectively. We assume τ₁, τ₂ to be piecewise monotonic and preserving continuous invariant measures μ₁ and μ₂, respectively. Let % F⁽¹⁾ and F⁽²⁾ be the distribution functions of μ₁ and μ₂. 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 τ₁ and τ₂ (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.