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

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.