度量空间的计盒测度
Box-counting measure of metric spaces
摘要：
本文引入了度量空间的\emph{box-counting measure}概念。我们证明了在双倍度量空间中，Ahlfors正则测度总是计盒测度;因此，如果$E$是满足开集条件的自相似集，则限制于$E$的Hausdorff测度是计盒测度.我们证明了两类自仿射集，广义Lalley-Gatzouras型自仿射海绵和Baraubrski地毯，总是允许盒计数措施，这也提供了一个非常简单的方法来计算这些分形盒维数。此外，除其他外，我们表明，如果两个加倍度量空间允许盒计数措施，那么多重分形谱的盒计数措施重合的两个空间是Lipschitz等价的。
英文摘要：
In this paper, we introduce a new notion called the \emph{box-counting measure} of a metric space. We show that for a doubling metric space, an Ahlfors regular measure is always a box-counting measure; consequently, if $E$ is a self-similar set satisfying the open set condition, then the Hausdorff measure restricted to $E$ is a box-counting measure. We show two classes of self-affine sets, the generalized Lalley-Gatzouras type self-affine sponges and Barański carpets, always admit box-counting measures; this also provides a very simple method to calculate the box-dimension of these fractals. Moreover, among others, we show that if two doubling metric spaces admit box-counting measures, then the multi-fractal spectra of the box-counting measures coincide provided the two spaces are Lipschitz equivalent.
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