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

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

2007-07-05

arxiv.org/abs/0707.0840v1

Mathematics

Functional Analysis

16 pages

Scientific paper

10.1007/s00220-008-0673-4

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.

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