N-Tupling Transformations and Invariant Definite Integrals

Functions on the real number line of the type ψ(x) = c + x − b/ x−μ, with b 0, have the interesting property that for any continuous, absolutely integrable function F on R, the graph of F(ψ(x)) is a “doubling” of the graph of F(x), while the integral over R remains invariant, ʃ∞−∞ F(ψ(x)) dx = ʃ∞−∞ F(x) dx. In this paper, we discover new families of n-to-1 mappings on R that have the same invariance property.