Derivations and Dirichlet forms on fractals

作者：Ionescu Marius; Rogers Luke G; Teplyaev Alexander^*

来源：Journal of Functional Analysis, 2012, 263(8): 2141-2169.

DOI：10.1016/j.jfa.2012.05.021

• Fredholm module
• Derivation
• Metric space
• Dirichlet form
• Finitely ramified fractal

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 non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar ease we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.

• 出版日期2012-10-15