论文标题
海绵上的自我措施措施的补充维度
The Assouad dimension of self-affine measures on sponges
论文作者
论文摘要
我们在$ \ mathbb {r}^d $中得出了自携带度量的上限和下限，并通过对角矩阵产生并满足适当的分离条件。上限和下限总是重合$ d = 2,3 $为尺寸提供精确的显式公式。此外，可以轻松检查条件，确保界限重合$ d \ geq 4 $。 我们结果的一个有趣的结果是，即使在平面中，这种自我处理结构也可能存在“维度差距”。也就是说，我们表明，对于“Barański型”的某些自动地毯，所有相关的自我措施的Assouad维度严格超过了地毯的Assouad尺寸，而某些固定的$δ> 0 $仅取决于地毯。我们还提供了没有尺寸差距的“Barański型”的自植物地毯的例子，实际上，地毯的Assouad尺寸等于精心选择的自我承包措施的Assouad尺寸。
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.
