A constructive proof of the existence of a semi-conjugacy for a one dimensional map

A continuous map $f:[0,1]\rightarrow[0,1]$ is called an $n$-modalmap if there is a partition $0=z_0 < z_1 < ... < z_n=1$ such that$f(z_{2i})=0$, $f(z_{2i+1})=1$ and, $f$ is (not necessarily strictly)monotone on each $[z_{i},z_{i+1}]$. It is well-known that such amap is topologically semi-conjugate to a piecewise linear map; howeverhere we prove that the topological semi-conjugacy is unique for thisclass of maps; also our proof is constructive and yields a sequenceof easily computable piecewise linear maps which converges uniformlyto the semi-conjugacy. We also give equivalent conditions for thesemi-conjugacy to be a conjugacy as in Parry's theorem. Related workwas done by Fotiades and Boudourides and Banks, Dragan and Jones,who however only considered cases where a conjugacy exists. Banks,Dragan and Jones gave an algorithm to construct the conjugacy mapbut only for one-hump maps.