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arXiv · 2407.16699

Fourier Decay from $L^2$-Flattening

Simon Baker·Osama Khalil·Tuomas Sahlsten
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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.

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Simon Baker, Osama Khalil, Tuomas Sahlsten. 2024-07-01. Fourier Decay from $L^2$-Flattening. https://arxiv.org/abs/2407.16699

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