Piecewise linear unimodal maps with non-trivial continuous piecewise linear commutator

Abstract

Let $g:[0,1]\to [0,1]$ be piecewise linear unimodal map. We say that $g$ has non-trivial piecewise linear commutator, if there is a continuous piecewise linear $\psi :[0,1]\to [0,1]$ such that $g\circ \psi =\psi \circ g$, and, moreover, $\psi$ is neither an iteration of $g$, not a constant map. We prove that if $g$ has a non-trivial piecewise linear commutator, then $g$ is topologically conjugated with the tent map by a piecewise linear conjugacy.

17 pages, 3 figures