A one-parameter family of unimodal, concave, polynomial maps of the interval exhibiting multiplicities

This paper shows there exists a polynomial map, p, of the interval 0, 1] onto itself that is concave, symmetric about the point and such that, when parameterized {μp}, 0 ≤ μ ≤ 1, there exist three distinct values of the parameter μ₀ < μ₁ < μ₂ such that RLR³C = K(μ₀p) ≠ K(μ₁p) ≠ K(μ₂p) = RLR³C. There is also given an explicit construction of a C¹ family with the same properties.