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

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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 (d S , ∞)–summable, the summability exponent d S 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 d S -energy functional which is shown to be a self-similar conformal invariant.

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Fabio Cipriani

  2. Institut de Mathématiques, CNRS-Université Pierre et Marie Curie, Boite 191, 4 Place Jussieu, F-75252, Paris Cedex 05, France

    Jean-Luc Sauvageot

Authors
  1. Fabio Cipriani
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  2. Jean-Luc Sauvageot
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Correspondence to Fabio Cipriani.

Additional information

Communicated by A. Connes

Thiswork has been supported by the project “Teoria ellittica e forme di Dirichlet su spazi frattali” G.N.A.M.P.A. 2008 and by the G.R.E.F.I.-G.E.N.C.O. French-Italian Research Group.

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Cipriani, F., Sauvageot, JL. Fredholm Modules on P.C.F. Self-Similar Fractals and Their Conformal Geometry. Commun. Math. Phys. 286, 541–558 (2009). https://doi.org/10.1007/s00220-008-0673-4

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  • DOI: https://doi.org/10.1007/s00220-008-0673-4

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