Completely symmetric resistance forms on the stretched Sierpiński gasket

Abstract

The stretched Sierpiński gasket, SSGfor short, is the space obtained by replacing every branching point of the Sierpiński gasket by an interval. It has also been called the “deformed Sierpiński 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 Sierpiński 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 Sierpiński gasket.