Fourier Decay from $L^2$-Flattening

TL;DR

This paper introduces a unified method leveraging $L^2$-flattening and derivative estimates to establish Fourier decay rates for various dynamically defined measures, impacting spectral gaps and fractal analysis.

Contribution

It presents a novel, systematic approach combining $L^2$-flattening with derivative estimates to derive Fourier decay for a broad class of measures.

Findings

Polylogarithmic decay for Diophantine self-similar measures

Polynomial decay for Patterson-Sullivan measures of hyperbolic manifolds

Applications to spectral gaps, fractal uncertainty, and equidistribution

Abstract

We develop a unified approach for establishing rates of decay for the Fourier transform of a wide class of dynamically defined measures. Among the key features of the method is the systematic use of the $L^2$-flattening theorem obtained in \cite{Khalil-Mixing}, coupled with non-concentration estimates for the derivatives of the underlying dynamical system. This method yields polylogarithmic Fourier decay for Diophantine self-similar measures, and polynomial decay for Patterson-Sullivan measures of convex cocompact hyperbolic manifolds, Gibbs measures associated to non-integrable $C^2$ conformal systems, as well as stationary measures for carpet-like non-conformal iterated function systems. Applications include essential spectral gaps on convex cocompact hyperbolic manifolds, fractal uncertainty principles, and equidistribution properties of typical vectors in fractal sets.