Non-degeneracy of the harmonic structure on Sierpinski Gaskets

TL;DR

This paper proves the invertibility of harmonic extension matrices for level-k Sierpinski Gaskets for all k>2, confirming a conjecture and providing conditions for non-degeneracy in self-similar fractals.

Contribution

It establishes the invertibility of harmonic extension matrices for all levels of Sierpinski Gaskets and offers a necessary condition for non-degeneracy in general self-similar sets.

Findings

Harmonic extension matrices are invertible for all k>2

Confirmed Hino's conjecture through proof and numerical tests

Provided a vertex connectivity-based necessary condition for non-degeneracy

Abstract

We prove that the harmonic extension matrices for the level-k Sierpinski Gasket are invertible for every k>2. This has been previously conjectured to be true by Hino in [6] and [7] and tested numerically for k<50. We also give a necessary condition for the non-degeneracy of the harmonic structure for general finitely ramified self-similar sets based on the vertex connectivity of their first graph approximation.