Spectral triples for the Sierpinski Gasket

TL;DR

This paper constructs spectral triples for the Sierpinski Gasket, revealing its geometric and spectral properties, including dimension, measure, and Dirichlet form, and establishing their relations to classical metrics and $K$-theory.

Contribution

It introduces a family of spectral triples for the Sierpinski Gasket and analyzes their spectral, metric, and measure-theoretic properties, connecting noncommutative geometry with fractal analysis.

Findings

The spectral triples recover the Hausdorff measure via residue calculations.

The Connes' distance is bi-Lipschitz equivalent to the Euclidean metric.

The energy dimension is identified with a specific logarithmic ratio.

Abstract

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 $a\to$ 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 $\delta_D$ of the energy functional $a\to$ tr$(|D|^{-s/2}|[D,a]|^2\,|D|^{-s/2})$ takes the value $d_E=\frac{\log(12/5)}{\log 2}$, which we call energy dimension, and the corresponding residue gives the standard Dirichlet form on $K$.