Computation of the scaling factor of resistance forms of the pillow and fractalina fractals

TL;DR

This paper introduces two new fractals, the fractalina and pillow, and calculates their resistance scaling factors, advancing understanding of harmonic structures on self-similar fractals.

Contribution

It computes the resistance scaling factors for the fractalina and pillow fractals, providing new insights into their harmonic structures.

Findings

Fractalina has a resistance scaling factor of (3+√41)/16.

Pillow fractal has a resistance scaling factor of ∛2.

These factors enable the computation of harmonic structures on these fractals.

Abstract

Much is known in the analysis of a finitely ramified self-similar fractal when the fractal has a harmonic structure: a Dirichlet form which respects the self-similarity of a fractal. What is still an open question is when such structure exists in general. In this paper, we introduce two fractals, the fractalina and pillow, and compute their resistance scaling factor. This is the factor which dictates how the Dirichlet form scales with the self-similarity of the fractal. By knowing this factor one can compute the harmonic structure on the fractal. The fractalina has scaling factor $(3+\sqrt{41})/16$, and the pillow fractal has scaling factor $\sqrt[3]{2}$.