Fourier transform and expanding maps on Cantor sets

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

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