Fourier dimension of Mandelbrot multiplicative cascades
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913848330551296 |
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| author | Chen, Changhao Li, Bing Suomala, Ville |
| author_facet | Chen, Changhao Li, Bing Suomala, Ville |
| contents | We investigate the Fourier dimension, $\dim_Fμ$, of Mandelbrot multiplicative cascade measures $μ$ on the $d$-dimensional unit cube. We show that if $μ$ is the cascade measure generated by a sub-exponential random variable then \[\dim_Fμ=\min\{2,\dim_2μ\}\,,\] where $\dim_2μ$ is the correlation dimension of $μ$ and it has an explicit formula.
For cascades on the circle $S\subset\mathbb{R}^2$, we obtain \[\dim_Fμ\ge\frac{\dim_2μ}{2+\dim_2μ}\,.\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fourier dimension of Mandelbrot multiplicative cascades Chen, Changhao Li, Bing Suomala, Ville Probability Mathematical Physics Classical Analysis and ODEs 60G57 (Primary), 42B10, 28A80 (Secondary) We investigate the Fourier dimension, $\dim_Fμ$, of Mandelbrot multiplicative cascade measures $μ$ on the $d$-dimensional unit cube. We show that if $μ$ is the cascade measure generated by a sub-exponential random variable then \[\dim_Fμ=\min\{2,\dim_2μ\}\,,\] where $\dim_2μ$ is the correlation dimension of $μ$ and it has an explicit formula. For cascades on the circle $S\subset\mathbb{R}^2$, we obtain \[\dim_Fμ\ge\frac{\dim_2μ}{2+\dim_2μ}\,.\] |
| title | Fourier dimension of Mandelbrot multiplicative cascades |
| topic | Probability Mathematical Physics Classical Analysis and ODEs 60G57 (Primary), 42B10, 28A80 (Secondary) |
| url | https://arxiv.org/abs/2409.13455 |