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

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.