Journal of Fractal Geometry

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Volume 6, Issue 2, 2019, pp. 143–156
DOI: 10.4171/JFG/73

Published online: 2019-03-21

Non-degeneracy of the harmonic structure on Sierpiński gaskets

Konstantinos Tsougkas[1]

(1) Uppsala University, Sweden

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

Keywords: Harmonic structure, harmonic extension matrices, energy Laplacian, Sierpiński gasket, prefractal graphs

Tsougkas Konstantinos: Non-degeneracy of the harmonic structure on Sierpiński gaskets. J. Fractal Geom. 6 (2019), 143-156. doi: 10.4171/JFG/73