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

@article{Lapidus2014DiracOA, title={Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets}, author={Michel L. Lapidus and Jonathan J. Sarhad}, journal={Journal of Noncommutative Geometry}, year={2014}, volume={8}, pages={947-985} }

We construct Dirac operators and spectral triples for certain, not necessarily self-similar, 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-affine space with continuously differentiable geodesics. As a fractal analog of Connes' theorem for a compact Riemmanian manifold, it…

## 18 Citations

Spectral triples for the variants of the Sierpiński gasket

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A new asymmetric criterion for the Riemann hypothesis (RH) is established in terms of the invertibility of the spectral operator for all values of the dimension parameter (i.e. for all c in the left half of the critical interval (0,1)).

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It is shown that, for nested fractals [T.Lindstrom, Mem. Amer. Math. Soc. 420, 1990], the main structural data, such as the Hausdorff dimension and measure, the geodesic distance (when it exists)…

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back

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In this paper, we present two new ways to associate a spectral triple to a higher-rank graph $\Lambda$. Moreover, we prove that these spectral triples are intimately connected to the wavelet…

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AbstractThe theory of “zeta functions of fractal strings” has been initiated by the first author in the early 1990s and developed jointly with his collaborators during almost two decades of intensive…

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