A combinatorial Fredholm module on self-similar sets built on $n$-cubes

TL;DR

This paper constructs a higher-dimensional Fredholm module on self-similar fractal sets like the Cantor dust and Sierpinski carpet, extending Connes' combinatorial approach and calculating associated Dixmier traces.

Contribution

It introduces a novel higher-dimensional combinatorial Fredholm module on self-similar sets, generalizing previous one-dimensional constructions.

Findings

Constructed Fredholm modules on various self-similar fractals

Calculated Dixmier traces of operators induced by the modules

Extended Connes' combinatorial approach to higher dimensions

Abstract

We construct a Fredholm module on self-similar sets such as the Cantor dust, the Sierpinski carpet and the Menger sponge. Our construction is a higher dimensional analogue of Connes' combinatorial construction of the Fredholm module on the Cantor set. We also calculate the Dixmier trace of two operators induced by the Fredholm module.