GAPS IN THE RATIOS OF THE SPECTRA OF LAPLACIANS ON FRACTALS

In contrast to the classical situation, it is known that many Laplacian operators on fractals have gaps in their spectrum. This surprising fact means there can be no limit in the Weyl counting formula and it is a key ingredient in proving that the convergence of Fourier series on fractals can be better than in the classical setting. Recently, it was observed that the Laplacian on the Sierpinski gasket has the stronger property that there are intervals which contain no ratios of eigenvalues. In this paper we give general criteria for this phenomena and show that Laplacians on many interesting classes of fractals satisfy our criteria.

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Covering Maps and Periodic Functions on Higher Dimensional Sierpinski Gaskets

Canadian Journal of Mathematics ◽

10.4153/cjm-2009-054-5 ◽

2010 ◽

Vol 61 (5) ◽

pp. 1151-1181 ◽

Cited By ~ 1

Author(s):

Huo-Jun Ruan ◽

Robert S. Strichartz

Keyword(s):

Fourier Series ◽

Periodic Function ◽

Sierpinski Gasket ◽

Series Expansions ◽

Periodic Functions ◽

Sierpiński Gasket ◽

Higher Dimensional ◽

Sierpinski Gaskets

Abstract.We construct covering maps from infinite blowups of the$n$-dimensional Sierpinski gasket$S{{G}_{n}}$to certain compact fractafolds based on$S{{G}_{n}}$. These maps are fractal analogs of the usual covering maps fromthe line to the circle. The construction extends work of the second author in the case$n=2$, but a differentmethod of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.

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NUMERICAL ANALYSIS ON THE SIERPINSKI GASKET, WITH APPLICATIONS TO SCHRÖDINGER EQUATIONS, WAVE EQUATION, AND GIBBS' PHENOMENON

Fractals ◽

10.1142/s0218348x04002689 ◽

2004 ◽

Vol 12 (04) ◽

pp. 413-449 ◽

Cited By ~ 10

Author(s):

KEVIN COLETTA ◽

KEALEY DIAS ◽

ROBERT S. STRICHARTZ

Keyword(s):

Finite Element Method ◽

Wave Equation ◽

Fourier Series ◽

Sierpinski Gasket ◽

Gibbs Phenomenon ◽

The Finite Element Method ◽

Sierpiński Gasket ◽

Jump Discontinuities ◽

Self Similar ◽

Series Type

We show how to improve the finite element method on the Sierpinski gasket (SG) to allow arbitrary partitions of the space. We use this method to study numerically solutions of the Schrödinger equation with well-type potentials, and the wave equation. We also show that Fourier series-type expansions on SG of functions with jump discontinuities appear to exhibit a self-similar Gibbs' phenomenon.

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