Some aspects of Cantor sets

Cantor sets;cut-out sets;Hausdorff measures;packing measures;dimension partition;multifractal analysis;L^p improving;gauge functions;Pure Mathematics

Ng, Ka Shing

University of Waterloo

关键词: Cantor sets;    cut-out sets;    Hausdorff measures;    packing measures;    dimension partition;    multifractal analysis;    L^p improving;    gauge functions;    Pure Mathematics;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/8300/3/Ng_Ka_Shing.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

For every positive, decreasing, summable sequence $a=(a_i)$, we can construct a Cantor set $C_a$ associated with $a$. These Cantor sets are not necessarily self-similar. Their dimensional properties and measures have been studied in terms of the sequence $a$.In this thesis, we extend these results to a more general collection of Cantor sets. We study their Hausdorff and packing measures, and compare the size of Cantor sets with the more refined notion of dimension partitions. The properties of these Cantor sets in relation to the collection of cut-out sets are then considered. The multifractal spectrum of $mathbf{p}$-Cantor measures on these Cantor sets are also computed. We then focus on the special case of homogeneous Cantor sets and obtain a more accurate estimate of their exact measures. Finally, we prove the $L^p$-improving property of the $mathbf{p}$-Cantor measure on a homogeneous Cantor set as a convolution operator.