11 research outputs found
A combinatorial Fredholm module on self-similar sets built on -cubes
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.Comment: 24 page
A combinatorial integration on the Cantor dust
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
A combinatorial integration on the Cantor dust
In this paper, we generalize the Cantor function to -dimensional cubes and
construct a cyclic -cocycle on the Cantor dust. This cocycle is non-trivial
on the pullback of the smooth functions on the -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 -forms on
the torus by using a combinatorial Fredholm module.Comment: 8 page