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

TL;DR

The paper proves that certain piecewise linear unimodal maps with non-trivial commutators are topologically conjugate to the tent map via a piecewise linear conjugacy, revealing a structural classification.

Contribution

It establishes that such maps with non-trivial commutators are topologically conjugate to the tent map through a piecewise linear conjugacy, characterizing their dynamics.

Findings

Maps with non-trivial commutators are conjugate to the tent map.

The conjugacy is piecewise linear.

Non-trivial commutators imply a specific dynamical structure.

Abstract

Let $g:\, [0, 1]\rightarrow [0, 1]$ be piecewise linear unimodal map. We say that $g$ has non-trivial piecewise linear commutator, if there is a continuous piecewise linear $\psi:\, [0, 1]\rightarrow [0, 1]$ 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.