  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.
