Fourier transform and expanding maps on Cantor sets

TL;DR

This paper investigates the decay properties of Fourier transforms of Gibbs measures for expanding maps, showing polynomial decay under certain non-linear conditions, which advances understanding of harmonic analysis on fractal measures.

Contribution

It establishes polynomial decay of Fourier transforms for Gibbs measures on expanding maps with strong separation and non-linearity, extending previous results to more general settings.

Findings

Fourier transforms decay polynomially for non-linear expanding maps

Decay rate depends on the non-linearity of the map

Results apply to measures on Cantor sets with strong separation

Abstract

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 $|\xi| \to \infty$.