Spectral asymptotics of Laplacians associated with a class of higher-dimensional graph-directed self-similar measures

Sze Man Ngai, Yuanyuan Xie

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

The spectral dimension of a fractal Laplacian encodes important geometric, analytic, and measure-theoretic information. Unlike standard Laplacians on Euclidean spaces or Riemannian manifolds, the spectral dimension of fractal Laplacians are often non-integral and difficult to compute. The computation is much harder in higher-dimensions. In this paper, we set up a framework for computing the spectral dimension of the Laplacians defined by a class of graph-directed self-similar measures on ℝd (d 2) satisfying the graph open set condition. The main ingredients of this framework include a technique of Naimark and Solomyak and a vector-valued renewal theorem of Lau et al.

Original languageEnglish
Pages (from-to)5375-5398
Number of pages24
JournalNonlinearity
Volume34
Issue number8
DOIs
StatePublished - Aug 2021

Scopus Subject Areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics

Keywords

  • fractal
  • graph-directed iterated function system
  • spectral dimension
  • the graph open set condition

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