Completely Symmetric Resistance Forms on the Stretched Sierpinski Gasket

TL;DR

This paper characterizes all symmetric resistance forms on the stretched Sierpinski gasket, revealing their structure as combinations of interval Dirichlet forms and the standard Sierpinski gasket resistance form, highlighting the geometric and symmetric properties.

Contribution

It identifies all completely symmetric resistance forms on the stretched Sierpinski gasket, linking them to interval Dirichlet forms and the classical Sierpinski gasket resistance form.

Findings

Resistance forms are sums of Dirichlet integrals on intervals with specific weights.

Resistance forms include linear combinations of interval forms and the standard Sierpinski gasket form.

The structure reflects the geometric and symmetric properties of the SSG.

Abstract

The stretched Sierpinski gasket, SSG for short, is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. It has also been called "deformed Sierpinski gasket" or "Hanoi attractor". As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing of the Brownian motions on the intervals. In fact, there have been several works in this direction. There still remains, however, "reminiscence" of the Sierpinski gasket in the geometric structure of SSG and the same should therefore be expected for diffusions. This paper shows that this is the case. In this work, we identify all the completely symmetric resistance forms on SSG. A completely symmetric resistance form is a resistance form whose restriction to every contractive copy of SSG in itself is invariant under all geometrical symmetries of the copy, which constitute the symmetry group of the triangle. We prove that completely symmetric resistance forms on SSG can be sums of the Dirichlet integrals on the intervals with some particular weights, or a linear combination of a resistance form of the former kind and the standard resistance form on the Sierpinski gasket.