Vibration modes of 3n-gaskets and other fractals

Abstract

We study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals which includes 3n-gaskets. We consider such examples as the Sierpinski gasket, a non-p.c.f. analog of the Sierpinski gasket, the level-3 Sierpinski gasket, a fractal 3-tree, the hexagasket, and one dimensional fractals. We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly. We support our theoretical analysis by symbolic and numerical computations.

Criteria for Spectral Gaps of Laplacians on Fractals

“…3) The spectrum of the standard Laplacian on this fractal has also been obtained independently in [2] and [6].…”

The spectra of the Laplacians of fractal graphs not satisfying spectral decimation

“…The first few graphs in the resulting sequence are shown in Figure 1. Figure 1: The first few graphs, Γ (0) , Γ (1) , Γ (2) and Γ (3) , in the sequence associated with the pentagasket. The filled in vertices are the boundary vertices.…”

Electrical resistance of N-gasket fractal networks

“…Note that the values J k are important since they allow to compute so-called harmonic matrices, that is, transformations h → h •ψ i acting on the space of harmonic functions [Kigami 2001;Strichartz 2006;Teplyaev ≥ 2007].…”

“…This paper is part of a relatively new, but now well-established, field of analysis and probability on fractals, see [Kigami 1993;1994;Barlow 1998;Strichartz 1999a;Mosco 2002;Adams et al 2003;Stanley et al 2003;Meyers et al 2004;Teplyaev 2004;≥ 2007;Bajorin et al ≥ 2007], and references therein for a sample of mathematical literature on the analysis on fractals. Furthermore, many of the questions addressed here are related to the general Dirichlet form theory; for further information on this subject the reader can refer to the now classical books [Bouleau and Hirsch 1991;Fukushima et al 1994].…”