Topological conjugate piecewise linear unimodal mappings of an interval into itself

Abstract

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