Derivations and Dirichlet forms on fractals

TL;DR

This paper explores the relationship between derivations, Fredholm modules, and topology on fractals with Dirichlet forms, revealing that non-trivial Fredholm modules exist only on non-tree fractals and connecting spectral properties with fractal geometry.

Contribution

It establishes a topological criterion for the existence of non-trivial Fredholm modules on fractals and relates spectral dimension to summability and harmonic function behavior.

Findings

Non-trivial Fredholm modules exist iff the fractal is not a tree.

Relation between spectral dimension and Fredholm module summability.

Connection between Lyapunov exponents, pressure function, and p-summability.

Abstract

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, and 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 case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.