Spectral asymptotics of Laplacians related to one-dimensional graph-directed self-similar measures with overlaps

Sze-Man Ngai Key Laboratory of High Performance Computing & Stochastic Information Processing, College of Math. & Stat., Hunan Normal University, Changsha, Hunan, China; and Dept. of Mathematical Sciences, Georgia Southern University, Statesboro, Ga., U.S.A. Yuanyuan Xie School of Mathematics, Renmin University of China, Beijing, China

Analysis of PDEs mathscidoc:2203.03006

Arkiv for Matematik, 58, (2), 393-435, 2020.11
For the class of graph-directed self-similar measures on R, which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined by these measures. For the class of finitely ramified graph-directed self-similar sets, the spectral dimension of the associated Laplace operators has been obtained by Hambly and Nyberg [6]. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.
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@inproceedings{sze-man2020spectral,
  title={Spectral asymptotics of Laplacians related to one-dimensional graph-directed self-similar measures with overlaps},
  author={Sze-Man Ngai, and Yuanyuan Xie},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220311101914911996949},
  booktitle={Arkiv for Matematik},
  volume={58},
  number={2},
  pages={393-435},
  year={2020},
}
Sze-Man Ngai, and Yuanyuan Xie. Spectral asymptotics of Laplacians related to one-dimensional graph-directed self-similar measures with overlaps. 2020. Vol. 58. In Arkiv for Matematik. pp.393-435. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220311101914911996949.
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