Non-degeneracy of the harmonic structure on Sierpiński gaskets

Abstract

We prove that the harmonic extension matrices for the two dimensional level-$k$ Sierpiński gasket are invertible for every $k\ge 2$. This has been previously conjectured to be true by Hino in [10] and [11] and tested numerically for $k\le 50$. 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.