Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917708424019968 |
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| author | Algom, Amir Hertz, Federico Rodriguez Wang, Zhiren |
| author_facet | Algom, Amir Hertz, Federico Rodriguez Wang, Zhiren |
| contents | We show that every self conformal measure with respect to a $C^2 (\mathbb{R})$ IFS $Φ$ has polynomial Fourier decay under some mild and natural non-linearity conditions. In particular, every such measure has polynomial decay if $Φ$ is $C^ω(\mathbb{R})$ and contains a non-affine map.
A key ingredient in our argument is a cocycle version of Dolgopyat's method, that does not require the cylinder covering of the attractor to be a Markov partition. It is used to obtain spectral gap-type estimates for the transfer operator, which in turn imply a renewal theorem with an exponential error term in the spirit of Li (2022). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_01275 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures Algom, Amir Hertz, Federico Rodriguez Wang, Zhiren Dynamical Systems Classical Analysis and ODEs We show that every self conformal measure with respect to a $C^2 (\mathbb{R})$ IFS $Φ$ has polynomial Fourier decay under some mild and natural non-linearity conditions. In particular, every such measure has polynomial decay if $Φ$ is $C^ω(\mathbb{R})$ and contains a non-affine map. A key ingredient in our argument is a cocycle version of Dolgopyat's method, that does not require the cylinder covering of the attractor to be a Markov partition. It is used to obtain spectral gap-type estimates for the transfer operator, which in turn imply a renewal theorem with an exponential error term in the spirit of Li (2022). |
| title | Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures |
| topic | Dynamical Systems Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2306.01275 |