Spectral triples and the geometry of fractals

Abstract

We construct spectral triples for the Sierpinski gasket as infinite sums of unbounded Fredholm modules associated with the holes in the gasket and investigate their properties. For each element in the K-homology group we find a representative induced by one of our spectral triples. Not all of these triples, however, will have the right geometric properties. If we want the metric induced by the spectral triple to give the geodesic distance, then we will have to include a certain minimal family of unbounded Fredholm modules. If we want the eigenvalues of the associated generalized Dirac operator to have the right summability properties, then we get limitations on the number of summands that can be included. If we want the Dixmier trace of the spectral triple to coincide with a multiple of the Hausdorff measure, then we must impose conditions on the distribution of the summands over the gasket. For the elements of a large subclass of the K-homology group, however, the representatives are induced by triples having the desired geometric properties. We finally show that the same techniques can be applied to the Sierpinski pyramid.