Abstract
We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in Rd 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≥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.
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.
| Original language | English |
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| Publication status | Submitted - 21 Mar 2022 |
Keywords
- Assouad type dimensions
- Self-affine set
- Dimension gap