Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets
type: Item
instance of: scholarly article
title: Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets
zbMATH Open document ID: 1181.46056
DOI: 10.4171/JNCG/43
author: Yanyan Li
published in: Journal of Noncommutative Geometry
publication date: 2009-07-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/0802.1336
review text: This article aims to define a noncommutative Riemannian structure on an ultrametric Cantor set \(\left( C,d\right) \). Predominant in the article is the view that ``Cantor sets should be treated as the boundary of a tree.'' Following [\textit{G. Michon}, C. R. Acad. Sci., Paris, Sér. I 300, 673--675 (1985; Zbl 0582.54019)], an ultrametric Cantor set \(\left( C,d\right) \) gives rise to a weighted rooted tree graph \(\mathcal{T}\left( C,d\right) \) with a boundary isometrically equivalent to \(C\) yielding a correspondence between a certain class of weighted rooted trees and ultrametrics on a Cantor set. The article grew out of the authors' desire to create a spectral triple for the transversal, \(\Xi\), of an aperiodic Delone set of finite type ([\textit{J. Bellissard}, Proceedings of the summer school on geometric and topological methods for quantum field theory, Villa de Leyva, Colombia, July 9--27, 2001. River Edge, NJ: World Scientific 86--156 (2003; Zbl 1055.81034)], \textit{J. Bellissard, R. Benedetti} and \textit{J.-M. Gambaudo} [Commun. Math. Phys. 261, No. 1, 1--41 (2006; Zbl 1116.46063)] and \textit{J. Bellissard, D. J. L. Herrmann} and \textit{M. Zarrouati} [Providence, RI: AMS, American Mathematical Society. CRM Monogr. Ser. 13, 207--258 (2000; Zbl 0972.52014)]), though it is not clear \(\Xi\)\ is embeddable in \(\mathbb{R}^{d}\) for any \(d\in\mathbb{N}^{+}\). The authors follow a proposal [\textit{A. Connes}, Noncommutative geometry. San Diego, CA: Academic Press. (1994; Zbl 0818.46076), Chap. IV.3. \(\varepsilon\)] for the triadic Cantor set, while Connes himself used his formalism to describe some properties of the Julia set there. In [Contemp. Math. 208, 211--252 (1997; Zbl 0889.58012)] \textit{M. L. Lapidus} proposed a program for creating spectral triples on fractals that would recapture both the geometric properties (fractal dimensions, Hausdorff measures, etc.) and the analytic properties (namely Laplacians) of the fractal at the same time. In […
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 53B21
Mathematics Subject Classification ID: 81R60
zbMATH DE Number: 5578953
zbMATH Keywords: Cantor sets
zbMATH Keywords: ultrametric
zbMATH Keywords: quantized calculus
zbMATH Keywords: zeta-functions
zbMATH Keywords: fractal exponents
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Lapacian, Brownian motion
MaRDI profile type: Publication
Wikidata QID: Q115212136
arXiv ID: 0802.1336
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description: scientific article