Topological Conjugate Piecewise Linear Unimodal Mappings of an Interval Into Itself

Let f, g : [0, 1] → [0, 1] be a pair of continuous piecewise linear unimodal mappings and let f be defined as follows: f(x) = 2x for x ≤ 1/2 and f(x) = 2 − 2x for x > 1/2. Also let h : [0, 1] → [0, 1] be a piecewise continuously differentiable homeomorphism such that fh = hg. Then h is piecewise linear and the mapping f, together with the ascending (or descending) monotone part of g, completely determines g and h.