Gaps in the Ratios of the Spectra of Laplacians on Fractals

Hare, Kathryn E.; Zhou, Denglin

doi:10.1142/s0218348x0900451x

Cited by 7 publications

(7 citation statements)

References 12 publications

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“…Thus it is of great interest to know whether similar operators exist for products of fractals other than SG. In fact [15] shows that this is the case for VS 2 and VS 3 . Also [9] investigates this question for a variant of the SG type fractal.…”

Section: Ratio Gapsmentioning

confidence: 81%

Analysis of the Laplacian and spectral operators on the Vicsek set

Constantin

Strichartz

Wheeler

2010

CPAA

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We study the spectral decomposition of the Laplacian on a family of fractals VSn that includes the Vicsek set for n = 2, extending earlier research on the Sierpinski Gasket. We implement an algorithm [24] for spectral decimation of eigenfunctions of the Laplacian, and explicitly compute these eigenfunctions and some of their properties. We give an algorithm for computing inner products of eigenfunctions. We explicitly compute solutions to the heat equation and wave equation for Neumann boundary conditions. We study gaps in the ratios of eigenvalues and eigenvalue clusters. We give an explicit formula for the Green's function on VSn. Finally, we explain how the spectrum of the Laplacian on VSn converges as n → ∞ to the spectrum of the Laplacian on two crossed lines (the limit of the sets VSn.)

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Section: Ratio Gapsmentioning

confidence: 81%

Analysis of the Laplacian and spectral operators on the Vicsek set

Constantin

Strichartz

Wheeler

2010

CPAA

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“…for the appropriate domains, and are extended by the polynomial representation everywhere else. An interesting feature of this result is that the function R(z) is given by a rational function which has disconnected real Julia set (see [26,27]) because R(z) has poles in the convex hull of its Julia set. One implication of this is that the spectrum of the renormalized limit of ∆ n , ∆ has gaps in the sense of [70], which implies that the Fourier series of continuous functions converge uniformly on K.…”

Section: Remark 22 Note That the Spectral Dimensionmentioning

confidence: 99%

“…Recently there has been considerable interest in Laplace operators with exotic spectral properties. For instance, large spectral gaps imply uniformly convergent Fourier expansions on fractals, see [70], and oscillations in the spectrum, see [2,4,26,27,12,40,42,41,43,50, and references therein]. However there had not been any infinite families of fractals with complicated spectra which can be computed.…”

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confidence: 99%

Gaps in the spectrum of the Laplacian on $3N$-Gaskets

Kelleher¹,

Gupta²,

Margenot³

et al. 2015

CPAA

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This article develops analysis on fractal 3N-gaskets, a class of post-critically finite fractals which include the Sierpinski triangle for N = 1, specifically properties of the Laplacian ∆ on these gaskets. We first prove the existence of a self-similar geodesic metric on these gaskets, and prove heat kernel estimates for this Laplacian with respect to the geodesic metric. We also compute the elements of the method of spectral decimation, a technique used to determine the spectrum of post-critically finite fractals. Spectral decimation on these gaskets arises from more complicated dynamics than in previous examples, i.e. the functions involved are rational rather than polynomial. Due to the nature of these dynamics, we are able to show that there are gaps in the spectrum.

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“…Strichartz [36] showed that the existence of spectral gaps implies better convergence of Fourier series. Rigorous proofs for the existence of spectral gaps have been obtained for only a limited number of fractals, such as the Sierpiński gasket and the Vicsek set (see [9,19,38]).…”

Section: Introductionmentioning

confidence: 99%