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 non-trivial Fredholm modules on post critically finite fractals by regular harmonic structures (D, r). The modules are (dS,∞)–summable, the summability exponent dS coinciding with the spectral dimension of the generalized Laplacian operator associated with (D, r). The characteristic tools of the noncommutative infinitesimal calculus allow to define a dS-energy functional which is shown to be a self-similar conformal invariant.