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

## TL;DR

This paper classifies unimodal maps on the interval that commute with certain piecewise linear surjective maps, revealing structural properties and illustrating conjugacy with the tent map.

## Contribution

It provides a new classification of unimodal maps commuting with piecewise linear surjective maps, enhancing understanding of their topological conjugacy properties.

## Key findings

- Unimodal maps commuting with piecewise linear surjective maps are classified.
- Such maps are topologically conjugate to the tent map.
- The classification illustrates the conjugacy relationship explicitly.

## Abstract

We introduce here a classification of unimodal maps $[0, 1]\rightarrow [0, 1]$, which commute with piecewise linear surjective maps $[0, 1]\rightarrow [0, 1]$. 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.