Fourier transform and expanding maps on Cantor sets

Abstract

We study the Fourier transforms $\hat{\mu }(\xi )$ 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 $\hat{\mu }(\xi )\to 0$ at a polynomial rate as $|\xi |\to \infty$.

31 pages, this paper supersedes arXiv:1810.01378. v5: Revised version. To appear in Amer. J. Math