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

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 R d (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.

Krein-Feller operators on Riemannian manifolds and Minkowski spaces

“…These Laplacians, as well as their generalizations, have been studied extensively in connection with fractal geometry, such as existence of an orthonormal basis of eigenfunctions, spectral dimension and spectral asymptotics, eigenvalues and eigenfunctions, eigenvalue estimates, differential equations, nodal inverse problems, wave equations and wave speed, heat equation and heat kernel estimates, etc. (see [4,7,12,16,[20][21][22][23][24][25]32,35,36,50,51,[53][54][55][56][57]62,63,69] and references therein). We remark that Freiberg and her coauthors defined a class of Laplacians that are more general in that Lebesgue measure is replaced by a more general measure (see, e.g., [21,23]).…”

Spectral dimensions of Krein-Feller operators in higher dimensions

“…The spectral dimension of Kreȋn-Feller operator for higher dimensions has been first computed by Triebel [Tri97,Theorem 30.2] for Ahlfors-David regular measures, by Naimark and Solomyak [Sol94;NS95] in the setting of self-similar measures under the open set condition (OSC), and recently by Ngai and Xie [NX21] for a class of graph-directed self-similar measures satisfying the graph open set condition. As an application of our general results we extend these achievements to self-conformal measures without any restriction on the separation conditions.…”