Cyclic Cohomology Groups of Some Self-similar Sets

TL;DR

This paper introduces a new form of Young integration tailored for cellular self-similar sets, demonstrating it as a cyclic 1-cocycle, thus linking it to the concept of currents in a novel setting.

Contribution

It develops a variant of Young integration for cellular self-similar sets and establishes its role as a cyclic 1-cocycle on the algebra of H"older continuous functions.

Findings

The variant of Young integration is well-defined on cellular self-similar sets.

The integration acts as a cyclic 1-cocycle, connecting to the theory of currents.

Provides criteria for the applicability of the integration variant.

Abstract

We define a variant of the Young integration on some kinds of self-similar sets which are called cellular self-similar sets. This variant is an analogue of the Young integration defined on the unit interval. We give the criteria of the variant on cellular self-similar sets, and also show that the variant is a cyclic 1-cocycle of the algebra of complex-valued H\"older continuous functions on the cellular self-similar sets. This suggests that the cocycle is a variant of currents.