Spectral triples for the Sierpinski gasket

摘要

We construct a 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 volume functional a →tr(a|D|~(-s)) at its abscissa of convergence d_D, which coincides with the Hausdorff dimension d_H of the fractal. We determine the associated Connes' distance showing that it is bi-Lipschitz equivalent to the distance on K induced by the Euclidean metric of the plane, and show that the pairing of the associated Fredholm module with (odd) K-theory is non-trivial. When the parameters belong to a suitable range, the abscissa of convergence δ_D of the energy functional a →tr(|D)|~(- a/ 2) |[D, a] | 2|D)|~(- s/ 2)) takes the value d_E = log(12/5)/log 2, which we call energy dimension, and the corresponding residue gives the standard Dirichlet form on K.