Metric Approximations of Spectral Triples on the Sierpi\'nski Gasket and other fractal curves

TL;DR

This paper demonstrates that spectral triples on fractals like the Sierpiński gasket can be approximated as limits of spectral triples on simpler finite graph approximations, using the spectral propinquity.

Contribution

It establishes a rigorous framework for the convergence of spectral triples on fractals via the spectral propinquity, connecting fractal geometry with noncommutative geometry.

Findings

Spectral triples on fractals are limits of those on approximating graphs.

The spectral propinquity effectively measures convergence of spectral triples.

Results have implications for analysis on fractals and potential physical applications.

Abstract

Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure: for instance, the Sierpi\'nski is the limit of finite graphs consisting of various affine images of an equilateral triangle. It is thus natural to ask whether the spectral triples, constructed on a class of fractals called piecewise $C^1$-fractal curves, are indeed limits, in an appropriate sense, of spectral triples on the approximating sets. We answer this question affirmatively in this paper, where we use the spectral propinquity on the class of metric spectral triples, in order to formalize the sought-after convergence of spectral triples. Our results and methods are relevant to the study of analysis on fractals and have potential physical applications.