Intermediate dimensions
Intermediate dimensions
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中间尺寸
DOI：
--
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发表时间：
2018
影响因子：
0.8
通讯作者：
Tom Kempton
中科院分区：
文献类型：
--
作者：
K. Falconer;J. Fraser;Tom Kempton
我们引入了一个维度的连续体，它是我们熟悉的Hausdorff维度和盒子维度之间的中间维度。这是通过限制豪斯道夫维度定义中允许的封面的系列来实现的，方法是坚持以下条件：|U|≤|V|θ\DocumentClass[12pt]{Minimal}\usepackage{amsath}\usepackage{waysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemarin}{-69pt}\Begin{Document}$|U|\le|V|^\theta$\end{Document其中，θ∈[0，1]\DocumentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$$\theta\in[0，1]$\end{Document}是一个参数。因此，当θ=1\DocumentCLASS[12pt]{Minimum}\UsPack{amsath}\UsPack{wa ysym}\UsPack{amsFonts}\UsPack{amssymb}\Usepackage{amsbsy}\UsPack{upgreek}\setLong{\oddsidemargin}{-69pt}\Begin{Document}$$\theta=1$$\end{Document}仅允许使用相同大小的集合，并且我们恢复框维度时，当θ=0\DocumentClass[12pt]{Minimum}\Usepackage{amsath}\usepackage{wa ysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\theta=0$$\end{Document}时，没有任何限制，我们恢复了Hausdorff维度。我们研究了中间维的许多性质(作为θ的函数)，包括证明它在(0，1)上连续但不一定在0处连续，并建立了质量分布原理、弗罗斯特曼引理、质量分布引理的适当类比以及产品的尺寸公式。我们还计算或估计了一些常见集合的中间维度，包括由整数的负幂组成的序列和Bedford-McMullen地毯。
We introduce a continuum of dimensions which are ‘intermediate’ between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that |U|≤|V|θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|U| \le |V|^\theta $$\end{document} for all sets U, V used in a particular cover, where θ∈[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in [0,1]$$\end{document} is a parameter. Thus, when θ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =1$$\end{document} only covers using sets of the same size are allowable, and we recover the box dimensions, and when θ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =0$$\end{document} there are no restrictions, and we recover Hausdorff dimension. We investigate many properties of the intermediate dimension (as a function of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}), including proving that it is continuous on (0, 1] but not necessarily continuous at 0, as well as establishing appropriate analogues of the mass distribution principle, Frostman’s lemma, and the dimension formulae for products. We also compute, or estimate, the intermediate dimensions of some familiar sets, including sequences formed by negative powers of integers, and Bedford–McMullen carpets.