The Assouad dimension of self-affine measures on sponges
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866911801717817344 |
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| author | Fraser, Jonathan M. Kolossváry, István |
| author_facet | Fraser, Jonathan M. Kolossváry, István |
| contents | We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\mathbb{R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$ yielding precise explicit formulae for the dimensions. Moreover, there are easy to check conditions guaranteeing that the bounds coincide for $d \geq 4$.
An interesting consequence of our results is that there can be a `dimension gap' for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of `Barański type' the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $δ>0$ depending only on the carpet. We also provide examples of self-affine carpets of `Barański type' where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_11247 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Assouad dimension of self-affine measures on sponges Fraser, Jonathan M. Kolossváry, István Dynamical Systems Classical Analysis and ODEs 28A80 (Primary) 37D20, 37C45 (Secondary) We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\mathbb{R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$ yielding precise explicit formulae for the dimensions. Moreover, there are easy to check conditions guaranteeing that the bounds coincide for $d \geq 4$. An interesting consequence of our results is that there can be a `dimension gap' for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of `Barański type' the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $δ>0$ depending only on the carpet. We also provide examples of self-affine carpets of `Barański type' where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure. |
| title | The Assouad dimension of self-affine measures on sponges |
| topic | Dynamical Systems Classical Analysis and ODEs 28A80 (Primary) 37D20, 37C45 (Secondary) |
| url | https://arxiv.org/abs/2203.11247 |