Journal of Noncommutative Geometry

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Volume 8, Issue 4, 2014, pp. 947–985
DOI: 10.4171/JNCG/174

Published online: 2015-02-02

Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets

Michel L. Lapidus[1] and Jonathan J. Sarhad[2]

(1) University of California, Riverside, USA
(2) University of California, Riverside, USA

We construct Dirac operators and spectral triples for certain, not necessarily selfsimilar, fractal sets built on curves. Connes’ distance formula of noncommutative geometry provides a natural metric on the fractal. To motivate the construction, we address Kigami’s measurable Riemannian geometry, which is a metric realization of the Sierpinski gasket as a self-ane space with continuously di erentiable geodesics. As a fractal analog of Connes’ theorem for a compact Riemmanian manifold, it is proved that the natural metric coincides with Kigami’s geodesic metric. This present work extends to the harmonic gasket and other fractals built on curves a significant part of the earlier results of E. Christensen, C. Ivan, and the first author obtained, in particular, for the Euclidean Sierpinski gasket. (As is now well known, the harmonic gasket, unlike the Euclidean gasket, is ideally suited to analysis on fractals. It can be viewed as the Euclidean gasket in harmonic coordinates.) Our current, broader framework allows for a variety of potential applications to geometric analysis on fractal manifolds.

Keywords: Analysis on fractals, noncommutative fractal geometry, Laplacians and Dirac operators on fractals, spectral triples, spectral dimension, measurable Riemannian geometry, geodesics on fractals, geodesic and noncommutative metrics, fractals built on curves, Euclidean and harmonic Sierpinski gaskets, geometric analysis on fractals, fractal manifold

Lapidus Michel, Sarhad Jonathan: Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets. J. Noncommut. Geom. 8 (2014), 947-985. doi: 10.4171/JNCG/174