Spectral triples for nested fractals

TL;DR

This paper demonstrates that spectral triples can effectively recover key geometric and analytic properties of nested fractals, providing a new framework for understanding their structure.

Contribution

It introduces a spectral triple approach to nested fractals, enabling the recovery of Hausdorff dimension, measure, geodesic distance, and energy from spectral data.

Findings

Spectral triples determine Hausdorff dimension and measure.

Spectral triples recover geodesic distance when it exists.

Spectral triples encode self-similar energy of fractals.

Abstract

It is shown that, for nested fractals [T.Lindstrom, Mem. Amer. Math. Soc. 420, 1990], the main structural data, such as the Hausdorff dimension and measure, the geodesic distance (when it exists) induced by the immersion in $R^n$, and the self-similar energy can all be recovered by the description of the fractals in terms of the spectral triples considered in [D.Guido, T.Isola, in "Advances in Operator Algebras and Mathematical Physics", Theta Series in Advanced Mathematics, Bucharest 2005].