用Peano曲线构造分形上的Laplacian
Using Peano Curves to Construct Laplacians on Fractals
摘要：
我们描述了一种新的方法来构建拉普拉斯分形使用Peano曲线从圆到分形，扩展的想法，已被用于某些朱利亚集的情况下。Peano曲线允许我们通过绘制回拉到圆的图形来可视化拉普拉斯算子的本征函数。我们详细研究了三种分形：五角形，八角形和魔毯。我们还使用的方法的两个非分形自相似集，环面和等边三角形，获得有吸引力的新的可视化特征函数的三角形。与许多熟悉的标准皮亚诺曲线近似的图片，不显示自相交，我们的描述近似的皮亚诺曲线有自相交，发挥了至关重要的作用，在构建图形近似的分形与明确的图形拉普拉斯算子，使分形拉普拉斯算子的限制。
英文摘要：
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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