Abstract
For the class of graph-directed self-similar measures on $\mathbf{R}$, 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.
Funding Statement
The authors are supported in part by the National Natural Science Foundation of China, grant 11771136, and Construct Program of the Key Discipline in Hunan Province. The first author is also supported in part by the Hunan Province Hundred Talents Program and a Faculty Research Scholarly Pursuit Funding from Georgia Southern University.
Citation
Sze-Man Ngai. Yuanyuan Xie. "Spectral asymptotics of Laplacians related to one-dimensional graph-directed self-similar measures with overlaps." Ark. Mat. 58 (2) 393 - 435, October 2020. https://doi.org/10.4310/ARKIV.2020.v58.n2.a9
Information
