Doubling Transformations and Definite Integrals

A doubling transformation ψ(x) = x − λ − b /x−μ, b, λ, μ ∈ R, b 0, has the property that for any absolutely integral function F(x) on R we have ʃ∞−∞ F(ψ(x)) dx = ʃ∞−∞ F(x) dx. Compositions of doubling transformations also satisfy this integral invariance property. In this paper we give criteria for determining when a given rational function is acomposition of two or more doubling transformations, and use this criteria for giving explicit families of such transformations such as [Mathematical formula].