Self-semiconjugation of piecewise linear unimodal maps

TL;DR

This paper investigates the structure of self-semiconjugations of piecewise linear unimodal maps, showing conditions under which all such functions are piecewise linear and characterizing the conjugacy to the tent map.

Contribution

It establishes criteria for when self-semiconjugations are piecewise linear and links this to the nature of conjugacy between the map and the tent map.

Findings

All self-semiconjugations are piecewise linear under certain conditions.

Conjugacy to the tent map is piecewise linear if all self-semiconjugations are piecewise linear.

Conditions involving the tangent at zero and the pre-image of zero determine the structure of self-semiconjugations.

Abstract

We devote this work to the functional equation $\psi \circ g = g\circ \psi$, where $\psi$ is an unknown function and $g$ is piecewise linear unimodal map, which is topologically conjugated to the tent map. We will call such $\psi$ self-semiconjugations of $g$. Our the main results are the following: 1. Suppose that there is a self-semiconjugation of $g$, whose tangent at $0$ is not a power of $2$, and suppose that all the kinks of $g$ are in the complete pre-image of $0$. Then all the self-semiconjugations of $g$ are piecewise linear. 2. Suppose that all self-semiconjugations of $g$ are piecewise linear. Then the conjugacy of $g$ and the tent map is piecewise linear.