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 language | English |
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Pages (from-to) | 5375-5398 |
Number of pages | 24 |
Journal | Nonlinearity |
Volume | 34 |
Issue number | 8 |
DOIs | |
State | Published - 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