The derivative of the conjugacy for the pair of tent-like maps from an interval into itself

TL;DR

This paper studies the topological conjugacy of a special class of piecewise linear unimodal maps called firm carcass maps, proving they are all conjugate and characterizing the derivative of the conjugacy.

Contribution

It establishes that all firm carcass maps are topologically conjugate and characterizes the derivative of the conjugacy in terms of piecewise linear approximations.

Findings

All firm carcass maps are topologically conjugate.

The conjugacy is either piecewise linear or has length 2.

The derivative of the conjugacy can be analyzed via linear approximations.

Abstract

We consider in this article the properties of the topological conjugacy of the piecewise linear unimodal maps $g:\, [0,\, 1]\rightarrow [0,\, 1]$, all whose kinks belong to the complete pre-image of $0$. We call such maps firm carcass maps. We prove that every firm carcass maps $g_1$ and $g_2$ are topologically conjugated. For the conjugacy $h$ such that $h\circ g_1 = g_2\circ h$ we denote $\{ h_n, n\geq 1\}$ the piecewise linear approximations of $h$, whose graphs connect the points $\{ (x, h(x)),\ g_1^n(x)=0\}$. For any $x\in [0,\, 1]$ we reduce the question about the value of $h'(x)$ to the properties of the sequence $\{h_n'(x),\, n\geq 1\}$. We prove that each conjugacy of firm carcass maps either has the length 2, or is piecewise linear.