Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets
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 [\textit{E. Christensen, C. Ivan} and \textit{M. L. Lapidus}, Adv. Math. 217, No. 1, 42--78 (2008; Zbl 1133.28002)] Lapidus and his collaborators have succeeded in performing much of the program for the Sierpinski gasket. In [Math. Scand. 100, No. 1, 35--60 (2007; Zbl 1155.58003)] \textit{E. Christensen} and \textit{C. Ivan} have suggested spectral triples for compact metric spaces by gluing together spectral triples associated with pairs of points, though they were unable to recover precise geometric information about the original space. The spectral triple given in this article extends much of the previous work and intrinsically captures the appropriate fractal geometric information without being forced to embed \(\Xi \)\ in \(\mathbb{R}^{d}\). The spectral triple enables us to construct an analogue of the Laplacian on the Cantor set as well as that of Brownian motion thereby. The present article has succeeded in extending the work to date by providing precise asymptonic estimates as \(t\rightarrow0\).
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