Geometry and Laplacian on discrete magic carpets

Journal of fractal geometry: mathematics of fractals and related topics

article

Chunyin Siu¹  Robert S. Strichartz¹ 

[1] Cornell University

关键词: Fractal;    Sierpinski carpet;    magic carpets;    metric balls;    random walk;    Laplacian;    heat kernel;    wave propagator;    harmonic functions;    Weyl ratio;   

DOI  :  10.4171/jfg/129

来源: European Mathematical Society

【 摘 要 】

We study several variants of the classical Sierpinski carpet (SC) fractal. The main examples we call infinite magic carpets (IMC), obtained by taking an infinite blowup of a discrete graph approximation to SC and identifying edges using torus, Klein bottle or projective plane type identifications. We use both theoretical and experimental methods. We prove estimates for the size of metric balls that are close to optimal. We obtain numerical approximations to the spectrum of the graph Laplacian on IMC and to solutions of the associated differential equations: Laplace equation, heat equation and wave equation. We present evidence that the random walk on IMC is transient, and that the full spectral resolution of the Laplacian on IMC involves only continuous spectrum. This paper is a contribution to a general program of eliminating unwanted boundaries in the theory of analysis on fractals.

【 授权许可】

Unknown