Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets

TL;DR

This paper constructs Dirac operators and spectral triples for fractal sets built on curves, establishing a natural metric via noncommutative geometry and extending analysis tools to fractal manifolds like the harmonic Sierpinski gasket.

Contribution

It introduces a framework for Dirac operators on non-self-similar fractals, linking Connes' metric with Kigami's geodesic metric, and extends previous results to the harmonic gasket.

Findings

Connes' distance formula matches Kigami's geodesic metric on fractals.

Spectral triples are constructed for fractals built on curves.

The framework broadens analysis possibilities on fractal manifolds.

Abstract

We construct Dirac operators and spectral triples for certain, not necessarily self-similar, fractal sets built on curves. Connes' distance formula of noncommutative geometry provides a natural metric on the fractal. To motivate the construction, we address Kigami's measurable Riemannian geometry, which is a metric realization of the Sierpinski gasket as a self-affine space with continuously differentiable geodesics. As a fractal analog of Connes' theorem for a compact Riemmanian manifold, it is proved that the natural metric coincides with Kigami's geodesic metric. This present work extends to the harmonic gasket and other fractals built on curves a significant part of the earlier results of E. Christensen, C. Ivan, and the first author obtained, in particular, for the Euclidean Sierpinski gasket. (As is now well known, the harmonic gasket, unlike the Euclidean gasket, is ideally suited to analysis on fractals. It can be viewed as the Euclidean gasket in harmonic coordinates.) Our current, broader framework allows for a variety of potential applications to geometric analysis on fractal manifolds.