Spectral triples and the geometry of fractals

TL;DR

This paper constructs spectral triples for K-homology elements of the Sierpinski gasket, demonstrating how these triples can recover geometric properties like metric, dimension, and measure, with limitations on the elements used.

Contribution

It introduces a method to generate spectral triples for K-homology elements of fractals, linking noncommutative geometry to fractal geometry.

Findings

Spectral triples can recover metric, dimension, and Hausdorff measure of the gasket.

Certain K-homology elements are limited in their ability to produce geometric information.

The approach applies to a large subgroup of the K-homology group.

Abstract

For each K-homolgy element of the Sierpinski gasket we construct a spectral triple which will generate that element. We show that there must be certain limits on the choice of the K-homology element if the geometric properties of the gasket shall be recoverable from that spectral triple. For a big subgroup of the K-homology group we show that our spectral triples will recover the metric, the dimension and the Hausdorff measure on the gaket.