  We study the Fourier transforms $\widehat{\mu}(\xi)$ of non-atomic Gibbs
measures $\mu$ for uniformly expanding maps $T$ of bounded distortions on
$[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear,
then $\widehat{\mu}(\xi) \to 0$ at a polynomial rate as $|\xi| \to \infty$.
