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Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities

Sze-Man Ngai Georgia Southern University Wei Tang Hunan Normal University Yuanyuan Xie Hunan Normal University

Publications of CMSA of Harvard mathscidoc:1611.38002

2017.6
We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps satisfy certain "bounded measure type condition", which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian defined by self-similar measure satisfying this condition. For Laplacians defined by fractal measures with overlaps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by Lau, Wang and Chu.
Fractal, Laplacian, spectral dimension, self-similar measures with overlaps, bounded measure type condition
[ Download ] [ 2016-11-14 05:55:05 uploaded by smngai ] [ 105 downloads ] [ 0 comments ]
  • This is a revised version of the previous version. The original definition has been generalized.
@inproceedings{sze-man2017spectral,
  title={Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities},
  author={Sze-Man Ngai, Wei Tang, and Yuanyuan Xie},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161114055505358933613},
  year={2017},
}
Sze-Man Ngai, Wei Tang, and Yuanyuan Xie. Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities. 2017. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161114055505358933613.
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