Classification of commutative pairs of surjective maps of interval, one of which is unimodal

We introduce here a classification of unimodal maps $[0, 1]\rightarrow [0, 1]$,whichcommutewithpiecewiselinearsurjectivemaps$[0, 1]\rightarrow [0, 1]$. Remind that if continuous piecewise linear unimodal map $g$ commutes with a non-constant piecewise linear map $\psi$, which is not an iteration of $g$, then $g$ is topologically conjugated with the tent map by piecewise linear conjugacy. We use the obtained classification to illustrate the mentioned fact.