Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket

Abstract

We provide a definition of integral, along paths in the Sierpinski gasket K, for differential smooth 1-forms associated to the standard Dirichlet form K. We show how this tool can be used to study the potential theory on K. In particular, we prove: i) a de Rham reconstruction of a 1-form from its periods around lacunas in K; ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; iii) the existence of potentials of smooth 1-forms on a suitable covering space of K. We finally show that this framework provides versions of the de Rham duality theorem for the fractal K.

Some proofs have been clarified, reference to previous literature is now more accurate, 33 pages, 6 figures

References in corpus (2)

• Spectral analysis on infinite Sierpinski fractafolds
• Derivations and Dirichlet forms on fractals

Cited by in corpus (13)

• Dirac and magnetic Schrödinger operators on fractals
• Spectral triples for the Sierpinski Gasket
• Metrics and spectral triples for Dirichlet and resistance forms
• Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets
• A note on reflected Dirichlet forms
• Spectra of Magnetic Operators on the Diamond Lattice Fractal
• Hodge-de Rham Theory on Fractal Graphs and Fractals
• From non-symmetric particle systems to non-linear PDEs on fractals
• A Spectral Triple for a Solenoid Based on the Sierpinski Gasket
• The emergence of Noncommutative Potential Theory
• A noncommutative Sierpinski Gasket
• Every Continuum has a Compact Universal Cover
• Variations in noncommutative potential theory: finite energy states, potentials and multipliers