New Results on Old Spectral Triples for Fractals

Abstract

It is shown that many important features of nested fractals, such as the Hausdorff dimension and measure, the geodesic distance induced by the immersion in $\mathbb{R}^n$ (when it exists), and the self-similar energy can be recovered by the description of the fractal in terms of spectral triples. We describe in particular the case of the Vicsek square, showing that all self-similar energies can be described through a deformation of the square to a rhombus.

Mathematics Subject Classification (2010). Primary 58B34; Secondary 28A80, 58J42