A combinatorial integration on the Cantor dust

TL;DR

This paper extends the Cantor function to two dimensions, constructing a non-trivial cyclic 2-cocycle on the Cantor dust that interacts with smooth functions on the torus, using combinatorial methods.

Contribution

It introduces a generalized Cantor function in 2D and constructs a cyclic 2-cocycle on the Cantor dust, linking it to smooth functions on the torus.

Findings

The cyclic 2-cocycle is non-trivial on the pullback of smooth functions.

The cocycle vanishes on Lipschitz functions on the Cantor dust.

The cocycle is computed via integration of 2-forms on the torus.

Abstract

In this paper, we generalize the Cantor function to $2$-dimensional cubes and construct a cyclic $2$-cocycle on the Cantor dust. This cocycle is non-trivial on the pullback of the smooth functions on the $2$-dimensional torus with the generalized Cantor function while it vanishes on the Lipschitz functions on the Cantor dust. The cocycle is calculated through the integration of $2$-forms on the torus by using a combinatorial Fredholm module.