On bounded distortions of maps in the line

TL;DR

This paper demonstrates that two different definitions of bounded distortion for smooth expanding maps on the real line are not equivalent, highlighting subtle distinctions in their mathematical properties.

Contribution

It provides a specific example showing the difference between two notions of bounded distortion for $ ext{C}^1$ expanding maps in $ ext{R}$.

Findings

Two notions of bounded distortion are not equivalent.

An explicit example illustrating the difference.

Clarifies the mathematical understanding of distortion in expanding maps.

Abstract

We give an example illustrating that two notions of bounded distortion for $\mathcal C^1$ expanding maps in $\R$ are different.