Fredholm Modules on P.C.F. Self-Similar Fractals and their Conformal Geometry

TL;DR

This paper constructs Fredholm modules on P.C.F. self-similar fractals using Dirichlet form-based differential calculus, linking spectral dimension with conformal invariance in a noncommutative geometric framework.

Contribution

It introduces a method to build Fredholm modules on fractals via harmonic structures, connecting spectral properties with conformal invariants.

Findings

Fredholm modules are d-summable with d equal to the spectral dimension.

The d-energy functional is a self-similar conformal invariant.

The approach bridges noncommutative geometry and fractal analysis.

Abstract

The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are d-summable, the summability exponent d coinciding with the spectral dimension of the generalized laplacian operator associated with the regular harmonic structures. The characteristic tools of the noncommutative infinitesimal calculus allow to define a d-energy functional which is shown to be a self-similar conformal invariant.