Fourier Decay from $L^2$-Flattening
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912162410135552 |
|---|---|
| author | Baker, Simon Khalil, Osama Sahlsten, Tuomas |
| author_facet | Baker, Simon Khalil, Osama Sahlsten, Tuomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16699 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fourier Decay from $L^2$-Flattening Baker, Simon Khalil, Osama Sahlsten, Tuomas Dynamical Systems Classical Analysis and ODEs Spectral Theory 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. |
| title | Fourier Decay from $L^2$-Flattening |
| topic | Dynamical Systems Classical Analysis and ODEs Spectral Theory |
| url | https://arxiv.org/abs/2407.16699 |