On a conjecture of the paper Differentiability of the Conjugacy in the Hartman-Grobman Theorem

TL;DR

This paper demonstrates that smooth bump functions are unnecessary for constructing differentiable conjugations in the Hartman-Grobman theorem, refuting a previous conjecture and proposing alternative smooth maps called blid maps.

Contribution

It shows that smooth blid maps can replace bump functions in conjugation constructions and provides a method to construct such maps in spaces lacking bump functions.

Findings

Smooth blid maps can replace bump functions in conjugation construction.

The conjecture about differentiability of conjugacy is incorrect.

Constructed a blid map for space C^0[0,1] without bump functions.

Abstract

In this note we show that for the construction of differentiable conjugation, the assumption of the existence of smooth bump function is not necessary, and consequently the corresponding conjecture stated in the paper of W. Zhang, K. Lu and W. Zhang "Differentiability of the Conjugacy in the Hartman-Grobman Theorem" (\cite{ZLZ}) is incorrect. We show that instead of bump functions we can use smooth blid maps. We also propose a construction of the blid map for the space $X=C^0[0,1]$, which does not possess a smooth bump function.