Dimension and measure for generic continuous images
Dimension and measure for generic continuous images
复制标题
通用连续图像的尺寸和测量
DOI：
10.5186/aasfm.2013.3819
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
J. Hyde
中科院分区：
文献类型：
--
作者：
R. Balka;'. Farkas;J. Fraser;J. Hyde
我们考虑从任意不可数紧度量空间X到rn的连续函数组成的Banach空间。关键问题是f(X)的一般维数是多少？我们考虑了两种不同的方法来回答这个问题：贝尔分类和流行。在Baire类别设置中，我们证明了典型的包装和上盒维度尽可能大，n，但发现Hausdor，下盒和拓扑维度的行为相当微妙。实际上，它们通常等于n的最小值和X的拓扑维数。我们还研究了f(X)的典型Hausdor测度和包装测度，特别是给出了它们为零、正有限或无限的必要条件。将贝尔分类结果与流行情况下的结果进行比较是很有趣的。因此，我们还讨论了Dougherty关于f(X)的流行拓扑维数的结果，并给出了紧度量空间上实值连续函数图的流行维数的一些简单应用，使我们能够推广Bayart和Heurteaux最近的结果。
We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, X, into R n . The key question is 'what is the generic dimension of f(X)?' and we consider two dierent approaches to answering it: Baire category and prevalence. In the Baire category setting we prove that typically the packing and upper box dimensions are as large as possible, n, but find that the behaviour of the Hausdor, lower box and topological dimensions is considerably more subtle. In fact, they are typically equal to the minimum of n and the topological dimension of X. We also study the typical Hausdor and packing measures of f(X) and, in particular, give necessary and sucient conditions for them to be zero, positive and finite, or infinite. It is interesting to compare the Baire category results with results in the prevalence setting. As such we also discuss a result of Dougherty on the prevalent topological dimension of f(X) and give some simple applications concerning the prevalent dimensions of graphs of real-valued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.