自仿射集的测度及其相关密度
Measure of Self-Affine Sets and Associated Densities
摘要：
设$B$是一个$n\times n$实扩张矩阵，$\mathcal{D}$是$\mathbb{R}^n$的有限子集，且有$0\in\mathcal{D}$ .自仿射集$K=K(B,\mathcal{D})$是满足集值方程$BK=\displaystyle\bigcup_{d\in\mathcal{D}}(K+d)$的唯一紧集。在$\text{card}(\mathcal{D})=\lvert\det B\rvert,$的情况下，我们将$K(B,\mathcal{D})$的勒贝格测度与相关测度$μ=\lim\limits_{s\to\infty}\sum\limits_{\ell_0,\dotsc,\ell_{s-1}\in\mathcal{D}}δ_{\ell_0+B\ell_1+\dotsb+B^{s-1}\ell_{s-1}}.$的上伯林密度联系起来。另一方面，如果$\text{card}(\mathcal{D})<\lvert\det B\rvert$和$B$是相似矩阵，我们将Hausdorff测度$\mathcal{H}^s(K)$（其中$s$是$K$的相似维数）与测度$μ$的上密度的相应概念联系起来。
英文摘要：
Let $B$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$ with $0\in\mathcal{D}$. The self-affine set $K=K(B,\mathcal{D})$ is the unique compact set satisfying the set-valued equation $BK=\displaystyle\bigcup_{d\in\mathcal{D}}(K+d)$. In the case where $\text{card}(\mathcal{D})=\lvert\det B\rvert,$ we relate the Lebesgue measure of $K(B,\mathcal{D})$ to the upper Beurling density of the associated measure $μ=\lim\limits_{s\to\infty}\sum\limits_{\ell_0,\dotsc,\ell_{s-1}\in\mathcal{D}}δ_{\ell_0+B\ell_1+\dotsb+B^{s-1}\ell_{s-1}}.$ If, on the other hand, $\text{card}(\mathcal{D})<\lvert\det B\rvert$ and $B$ is a similarity matrix, we relate the Hausdorff measure $\mathcal{H}^s(K)$, where $s$ is the similarity dimension of $K$, to a corresponding notion of upper density for the measure $μ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
