On one generalization of skew tent maps

TL;DR

This paper extends the understanding of conjugacy properties in skew tent maps by analyzing the differentiability of conjugacy functions within a broader class of piecewise linear maps, linking differential properties to their approximations.

Contribution

It introduces a new class of piecewise linear maps with kinks in the pre-image of zero and studies their conjugacy properties, generalizing previous results.

Findings

Conjugacy differentiability relates to the limits of derivatives of approximations.

The introduced class of maps broadens the scope of conjugacy analysis.

The relation between differential properties and approximations is established.

Abstract

We generalize in this work the properties of the conjugacy of skew tent maps. It is known that the conjugacy $h$ from a skew tent map $g_1$ to $g_2$ is differentiable at a point $x^*$ if and only if there exists left and right limits $\lim\limits_{n\rightarrow \infty}h_n'(x^*-)$ and $\lim\limits_{n\rightarrow \infty}h_n'(x^*+)$, where $h_n$ is a piecewise linear function, which coincides with $h$ at $g_1^{-n}(0)$, and all whose kinks belong to $g_1^{-n}(0)$. The attempts to generalize this result to some reacher class of unimodal maps is natural. For this reason we introduce the class of piecewise linear maps, all whose kinks are in the complete pre-image of $0$ and study the relation of the differential properties of their conjugacy with ones of the mentioned approximation~$(h_n)_{n\geq 1}$.