Decomposition of Dirichlet Forms Associated to Unbounded Dirichlet Operators

Ko, Chul Ki

doi:10.4134/bkms.2009.46.2.347

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2012

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“…Besides the reference [CS], which we use extensively, previous relevant papers of Cipriani and Sauvageot are [CS02,CS03gfa]. Concerning the relation between Dirichlet forms and operator algebras, our work was also influenced by [Ko1,Ko2,Si05]. In our paper we use the classical theory of local Dirichlet forms, see [BouleauHirsch, FOT, RöcknerMa].…”

Section: Introductionmentioning

confidence: 99%

Derivations and Dirichlet forms on fractals

Ionescu

Rogers

Teplyaev

2012

Journal of Functional Analysis

121

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We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including nonself-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.

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Section: Introductionmentioning

confidence: 99%