Arkiv för Matematik

Volume 58 (2020)

Number 2

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

Pages: 393 – 435



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)


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.


fractal, spectral dimension, graph-directed self-similar measure, essentially of finite type

2010 Mathematics Subject Classification

Primary 28A80, 35P20. Secondary 35J05.

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.

Received 19 October 2018

Received revised 20 May 2020

Accepted 3 June 2020

Published 3 November 2020