Spectral Triples for the Variants of the Sierpinski Gasket

TL;DR

This paper applies noncommutative geometric tools, specifically spectral triples, to analyze fractal sets like the harmonic and stretched Sierpinski gaskets, recovering key geometric measures and dimensions.

Contribution

It demonstrates how spectral triples can be used to extract classical fractal measures and dimensions from noncommutative geometric frameworks.

Findings

Spectral triples recover the self-affine measure of the harmonic Sierpinski gasket.

Spectral triples determine the Hausdorff dimension and measure of the stretched Sierpinski gasket.

The approach bridges noncommutative geometry and fractal analysis.

Abstract

Fractal geometry is the study of sets which exhibit the same pattern at multiple scales. Developing tools to study these sets is of great interest. One step towards developing some of these tools is recognizing the duality between topological spaces and commutative $C^\ast$-algebras. When one lifts the commutativity axiom, one gets what are called noncommutative spaces and the study of noncommutative geometry. The tools built to study noncommutative spaces can in fact be used to study fractal sets. In what follows we will use the spectral triples of noncommutative geometry to describe various notions from fractal geometry. We focus on the fractal sets known as the harmonic Sierpinski gasket and the stretched Sierpinski gasket, and show that the spectral triples constructed by Christensen, Ivan, and Lapidus in 2008 and Lapidus and Sarhad in 2015, can recover the standard self-affine measure in the case of the harmonic Sierpinski gasket and the Hausdorff dimension, geodesic metric, and Hausdorff measure in the case of the stretched Sierpinski gasket.