Hölder等价下的最小豪斯多夫维数
Minimising Hausdorff Dimension under Hölder Equivalence
摘要：
我们研究与给定度量空间Hölder等价的空间的豪斯多夫维数的下界值;我们称之为双Hölder不变的“Hölder维数”。这个定义和我们的一些方法类似于在共形维数的研究中所使用的那些。本文证明了紧致二重度量空间的Hölder维数是由容量维数上界的.作为推论，我们得到了紧的局部自相似度量空间的Hölder维数与拓扑维数相等.在这个过程中，我们证明了任何紧的，加倍度量空间可以映射到希尔伯特空间，使映射是一个bi-Hölder同胚到它的图像和豪斯多夫维数的图像是任意接近原空间的容量维。我们提供的例子来说明我们的结果的清晰度。例如，一个例子表明对于非自相似空间，Hölder维数可以严格大于拓扑维数，另一个例子表明Hölder维数不需要达到。
英文摘要：
We study the infimal value of the Hausdorff dimension of spaces that are Hölder equivalent to a given metric space; we call this bi-Hölder-invariant "Hölder dimension". This definition and some of our methods are analogous to those used in the study of conformal dimension. We prove that Hölder dimension is bounded above by capacity dimension for compact, doubling metric spaces. As a corollary, we obtain that Hölder dimension is equal to topological dimension for compact, locally self-similar metric spaces. In the process, we show that any compact, doubling metric space can be mapped into Hilbert space so that the map is a bi-Hölder homeomorphism onto its image and the Hausdorff dimension of the image is arbitrarily close to the original space's capacity dimension. We provide examples to illustrate the sharpness of our results. For instance, one example shows Hölder dimension can be strictly greater than topological dimension for non-self-similar spaces, and another shows the Hölder dimension need not be attained.
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