Using Peano Curves to Construct Laplacians on Fractals

Abstract

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.

Geometry and Laplacian on Discrete Magic Carpets

“…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.…”

“…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.…”

Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions

“…The paper [53] provides another technique for proving the existence of Dirichlet forms on non-finitely ramified self-similar fractals, which estimates the parameter ρ by studying the Poincaré inequalities on the approximating graphs of the fractals. The long term motivation for our work comes from probability and analysis on fractals [5,13,14,61,64,66], vector analysis for Dirichlet forms [33-35, 37-39, 54], and especially from the works on the heat kernel estimates [3, 6, 7, 24, 27, 41, 42, 48-50, 55, 56, 67].…”

Simple Fractal Calculus from Fractal Arithmetic

“…In the Sierpiński case the bijection has a space-filling property reminiscent of Peano curves [24]. The very idea that there are links between Sierpiński-type fractals and spacefilling curves is not new, and was used by Molitor et al in [25] in their construction of Laplacians on fractals. However, in all other respects the approach from [25] is different from what we discuss below.…”

“…The very idea that there are links between Sierpiński-type fractals and spacefilling curves is not new, and was used by Molitor et al in [25] in their construction of Laplacians on fractals. However, in all other respects the approach from [25] is different from what we discuss below. The idea of employing a one-dimensional integration for finding higher dimensional integrals is known [26], but apparently has not been used in fractal contexts so far.…”