Spectral triples for the Sierpinski Gasket

Details Spectral triples for the Sierpinski Gasket Spectral triples for the Sierpinski Gasket

2011-12-29

arxiv.org/abs/1112.6401v2

Mathematics

Operator Algebras

40 pages, 9 figures. Some points have been corrected, others have been clarified

Scientific paper

We construct a 2-parameter family of spectral triples for the Sierpinski Gasket K. For suitable values of the parameters we determine the dimensional spectrum and recover the Hausdorff measure of K in terms of the residue of the functional a --> tr_omega (a |D|^{-s}) at the abscissa of convergence d, which coincides with the Hausdorff dimension of the fractal. We determine the associated Connes' distance showing that it is bi-Lipschitz equivalent to a suitable root of the Euclidean metric of the plane, and show that the pairing of the associated Fredholm module with (odd) K-theory is non-trivial. We recover also the unique, standard Dirichlet form on K, as the residue of the functional a --> tr_omega (|D|^{-s/2}|[D,a]|^2 |D|^{-s/2}) at the abscissa of convergence delta, which we call the energy dimension. The fact that the Hausdorff dimension differs from the energy dimension reflects the fact that on K energy and volume are distributed singularly.

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