Derivations and Dirichlet forms on fractals

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.

to appear in the Journal of Functional Analysis 2012

References in corpus (6)

• Two-sided estimates of heat kernels on metric measure spaces
• Stability of parabolic Harnack inequalities on metric measure spaces
• Spectral analysis on infinite Sierpinski fractafolds
• Dirac and magnetic Schrödinger operators on fractals
• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

Cited by in corpus (20)

• Spectral analysis on infinite Sierpinski fractafolds
• Dirac and magnetic Schrödinger operators on fractals
• Spectral triples for the Sierpinski Gasket
• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket
• Energy and Laplacian on Hanoi-type fractal quantum graphs
• A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy
• Fractal snowflake domain diffusion with boundary and interior drifts
• Finite energy coordinates and vector analysis on fractals
• Differential forms on Dirichlet spaces and Bakry-Émery estimates on metric graphs
• Spectra of Magnetic Operators on the Diamond Lattice Fractal
• Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket
• Canonical diffusions on the pattern spaces of aperiodic Delone sets
• Hodge-de Rham Theory on Fractal Graphs and Fractals
• From non-symmetric particle systems to non-linear PDEs on fractals
• Gaps in the spectrum of the Laplacian on $3N$-Gaskets
• A noncommutative Sierpinski Gasket
• On some topics of analysis on noncommutative spaces
• Differential forms for fractal subspaces and finite energy coordinates
• Dual graphs and modified Barlow--Bass resistance estimates for repeated barycentric subdivisions