Spectral triples for the Sierpinski gasket

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 vertical bar D vertical bar(-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 delta(D) of the energy functional a -> tr(vertical bar D vertical bar(-s/2)vertical bar[D,a]vertical bar(2)vertical bar D vertical bar(-s/2)) takes the value d(E) lotg2/5)/log 2, which we call energy dimension, and the corresponding residue gives the standard Dirichlet form on K.