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|D|−s)a→tr(a|D|−s) at its abscissa of convergence dDdD, which coincides with the Hausdorff dimension dHdH 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 δDδD of the energy functional a→tr(|D|−s/2|[D,a]|2|D|−s/2)a→tr(|D|−s/2|[D,a]|2|D|−s/2) takes the value dE=log(12/5)log2, which we call energy dimension, and the corresponding residue gives the standard Dirichlet form on K.