Cheeger's energy on the Harmonic Sierpinski Gasket

TL;DR

This paper provides an alternative proof that Cheeger's energy coincides with the natural Dirichlet form on the harmonic Sierpinski gasket, utilizing Lyapounov exponents rather than previous methods.

Contribution

It introduces a novel proof approach based on Lyapounov exponents to establish the equivalence of Cheeger's energy and the Dirichlet form on the harmonic Sierpinski gasket.

Findings

Cheeger's energy equals the natural Dirichlet form on the harmonic Sierpinski gasket.

The proof employs properties of the Lyapounov exponent.

The result confirms the compatibility of different analytical frameworks on fractals.

Abstract

Koskela and Zhou have proven that, on the harmonic Sierpinski gasket with Kusuoka's measure, the "natural" Dirichlet form coincides with Cheeger's energy. We give a different proof of this result, which uses the properties of the Lyapounov exponent of the gasket.