Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space
Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space
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DOI：
10.1007/s00039-016-0368-5
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发表时间：
2012-11
影响因子：
2.2
通讯作者：
C. Bishop;Hrant Hakobyan;Marshall Williams
中科院分区：
文献类型：
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作者：
C. Bishop;Hrant Hakobyan;Marshall Williams
我们证明了它是Ahlfors正则空间之间的拟对称映射，那么对于“几乎所有”有界Ahlfors正则集。另外，对于“几乎每一个”ahfors正则集，都有Loewner空间。用测度的Fuglede模对这些结果作了精确的描述。作为这些一般定理的推论，我们证明了它是，的拟共形映射，那么对于Lebesgue a.e，我们有。类似的结果也适用于卡诺群。对于平面拟共形映射，我们的一般估计表明，如果是ahlfors正则，，则某个分量最多有维数，我们构造了例子来证明这个界是明确的。此外，我们还证明了存在不包含可整流子弧的a维集合和平面拟共形映射。这些结果概括了Balogh等人（J Math Pures appll(2) 99:125-149, 2013）的工作，并回答了Balogh等人（J Math Pures appll(2) 99:125-149, 2013）和Capogna等人（度量空间中的映射理论）提出的问题。http://aimpl.org/mappingmetric, 2016)。
We show that ifis a quasisymmetric mapping between Ahlfors regular spaces, thenfor “almost every” bounded Ahlfors regular set. If additionally,andare Loewner spaces thenfor “almost every" Ahlfors regular set. The precise statements of these results are given in terms of Fuglede’s modulus of measures. As a corollary of these general theorems we show that ifis a quasiconformal map of,, then for Lebesgue a.e.we have. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that ifis Ahlfors-regular,, then some component ofhas dimension at most, and we construct examples to show this bound is sharp. In addition, we show there is a-dimensional setand planar quasiconformal mapsuch thatcontains no rectifiable sub-arcs. These results generalize work of Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and answer questions posed in Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and Capogna et al. (Mapping theory in metric spaces. http://aimpl.org/mappingmetric , 2016).