Spectral triples for the variants of the Sierpiński gasket
Spectral triples for the variants of the Sierpiński gasket
复制标题
Sierpiński 垫片变体的光谱三元组
DOI：
--
复制
发表时间：
2017
影响因子：
0.8
通讯作者：
Andrea Arauza Rivera
中科院分区：
文献类型：
--
作者：
Andrea Arauza Rivera
分形几何是研究在多个尺度上表现出相同图案的集合。开发工具来研究这些集合是非常有趣的。发展这些工具的一个步骤是认识到拓扑空间和交换$C^ast$-代数之间的对偶性。当人们举起交换性公理时，人们得到了所谓的非交换空间和非交换几何的研究。为研究非交换空间而建立的工具实际上也可以用来研究分形集。在下文中，我们将使用非交换几何的谱三元组来描述分形几何的各种概念。我们专注于被称为调和Sierpinski垫片和拉伸Sierpinski垫片的分形集，并表明Christensen，Ivan和Lapidus在2008年以及Lapidus和Sarhad在2015年构造的谱三元组可以恢复调和Sierpinski垫片和Hausdorff维数，测地度量，和Hausdorff测度的情况下拉伸Sierpinski垫片。
Fractal geometry is the study of sets which exhibit the same pattern at multiple scales. Developing tools to study these sets is of great interest. One step towards developing some of these tools is recognizing the duality between topological spaces and commutative $C^ast$-algebras. When one lifts the commutativity axiom, one gets what are called noncommutative spaces and the study of noncommutative geometry. The tools built to study noncommutative spaces can in fact be used to study fractal sets. In what follows we will use the spectral triples of noncommutative geometry to describe various notions from fractal geometry. We focus on the fractal sets known as the harmonic Sierpinski gasket and the stretched Sierpinski gasket, and show that the spectral triples constructed by Christensen, Ivan, and Lapidus in 2008 and Lapidus and Sarhad in 2015, can recover the standard self-affine measure in the case of the harmonic Sierpinski gasket and the Hausdorff dimension, geodesic metric, and Hausdorff measure in the case of the stretched Sierpinski gasket.