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

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.

arXiv comment16 pages

References in corpus (1)

• Dimensions and singular traces for spectral triples, with applications to fractals

Cited by in corpus (7)

• Derivations and Dirichlet forms on fractals
• Dirac and magnetic Schrödinger operators on fractals
• Spectral triples for the Sierpinski Gasket
• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket
• Fractal snowflake domain diffusion with boundary and interior drifts
• Finite energy coordinates and vector analysis on fractals