Fourier Decay from L²-Flattening

2024/07/01 by Baker, Simon, Khalil, Osama, Sahlsten, Tuomas · 3 citations
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2407.16699

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 L²-flattening theorem obtained in \citeKhalil-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 C² 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.

Cited by

• The dimension of well approximable numbers
• Fourier decay and absolute continuity for typical homogeneous self-similar measures in \mathbb R^d for d≥ 3
• On the Fourier transform of random Bernoulli convolutions

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