Hodge-de Rham Theory of K-Forms on Carpet Type Fractals

Bello, Jason; Li, Yiran; Strichartz, Robert S.

doi:10.1007/978-3-319-13230-3_2

Cited by 4 publications

(8 citation statements)

References 17 publications

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“…Since this is greater than one, it would imply that points have zero capacity and functions of finite energy do not have to be continuous, in contrast to all PCF fractals, SC and PG. We also observe that the spectral data agrees exactly with the data in [3] for the approximations to the zero-forms Laplacian on MC. In fact the approximate graph Laplacians are identical.…”

Section: The Magic Carpetsupporting

confidence: 73%

“…The third case occurs exactly when x is a singular point introduced at level m, and the second case occurs when x is a singular point introduced at level m < m. We note that these approximate Laplacians agree exactly (except for a different renormalization constant) with the approximate Laplacians for zero-forms studied in [3], and indeed the eigenvalues shown in Table 5.1 are equal to six times the eigenvalues computed in [3]. We would like to believe that a limit as in (4.3) exists for an appropriate choice of r. Experimentally it appears that r ≈ 1.25 and 8 r ≈ 6.45.…”

Section: The Magic Carpetsupporting

confidence: 52%

“… 3. we show the log-log graphs of the eigenvalue counting function, ρ(x) = #{λ j ≤ x} and the Weyl Ratio, W R(x) = ρ(x)x β for β = log5 log5−log5 ≈ 0.675 on levels 2,3,and 4.…”

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confidence: 99%

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Using Peano Curves to Construct Laplacians on Fractals

2015

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We describe a new method to construct Laplacians on fractals using a Peano curve from the circle onto the fractal, extending an idea that has been used in the case of certain Julia sets. The Peano curve allows us to visualize eigenfunctions of the Laplacian by graphing the pullback to the circle. We study in detail three fractals: the pentagasket, the octagasket and the magic carpet. We also use the method for two nonfractal self-similar sets, the torus and the equilateral triangle, obtaining appealing new visualizations of eigenfunctions on the triangle. In contrast to the many familiar pictures of approximations to standard Peano curves, that do no show self-intersections, our descriptions of approximations to the Peano curves have self-intersections that play a vital role in constructing graph approximations to the fractal with explicit graph Laplacians that give the fractal Laplacian in the limit.

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Section: The Magic Carpetsupporting

confidence: 73%

Section: The Magic Carpetsupporting

confidence: 52%

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Using Peano Curves to Construct Laplacians on Fractals

2015

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“…on M C, for the appropriate renormalization factor R. Then the spectrum of −∆ would be the limit of the spectrum of ∆ (m) multiplied by R m . In fact, there is no published proof of the existence of this limit, but numerical data in this paper and in previous works ( [BLS15], [MOS15] in the torus identification case) leaves little doubt that the limit exists.…”

Section: Spectrum Of the Laplacian On Magic Carpet Fractalsmentioning

confidence: 55%

“…But a glance at Figure 1 shows that there are infinitely many line segments in SC that are locally isometric to portions of this boundary, so the standard choice appears somewhat arbitrary and capricious. In an attempt to get rid of the boundary altogether, a related fractal called the Magic Carpet (M C) was introduced in [BLS15] and further studied in [MOS15] where potential boundary line segments are identified. Thus the opposite sides of the original square are identified with the same orientation to produce a torus.…”

Section: Introductionmentioning

confidence: 99%

Geometry and Laplacian on Discrete Magic Carpets

Goodman¹,

Siu²,

Strichartz³

2019

Preprint

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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.

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Spectrum of the Laplacian on the Vicsek Set “With No Loose Ends”

Strichartz

Zhu

2017

Fractals

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We study the spectral properties of a fractal VNLE obtained from the standard Vicsek set VS by making a countable number of identifications of points so that all the line segments in VS become circles in VNLE. We show that the standard Laplacian on VNLE satisfies spectral decimation with the same cubic renormalization polynomial as for VS, and thereby give a complete description of all eigenfunctions of the Laplacian. We then study the restrictions of eigenfunctions to the large circles in VNLE and prove that these are Lipschitz functions.

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Hodge-de Rham Theory of K-Forms on Carpet Type Fractals

Cited by 4 publications

References 17 publications

Using Peano Curves to Construct Laplacians on Fractals

Using Peano Curves to Construct Laplacians on Fractals

Geometry and Laplacian on Discrete Magic Carpets

Spectrum of the Laplacian on the Vicsek Set “With No Loose Ends”

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