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.

Analysis on hybrid fractals

“…These hold for any graph-directed graph equipped with a self-similar energy. 4. Characterization condition of energy on finitely ramified cell structures.…”

“…The properties (RF1) and (RF4) follow immediately from the fact that F ⊆ F. The definition of F implies (RF2) and Lemma 4.7 yields (RF3). Finally, since V * is in particular R E -bounded, we can adapt [4,Theorem 11.6] to get (RF5).…”

“…Example (Hanoi attractor). This space, also called Stretched Sierpinski gasket [4], falls into the class of hybrids treated in Section 3. Its base is the Sierpinski gasket and the set of bonds consists of the three line segments that join each triangle in Figure 8.…”

“…In this regard, this paper addresses the following question: Does a resistance form reflect the intrinsic structure of the underlying space completely? Previous investigations [5,4] revealed the possibility that a resistance form may "miss" essential properties of the space on which it is defined, yielding an incomplete framework to treat analytic questions. More precisely, in the mentioned works a resistance form was constructed on the fractal set displayed in Figure 1 which did not produce "fractal analysis".…”

“…(ii) The latter sequence yields a unique resistance form (E, dom E) given by (4), For simplicity, we also denote by (E, dom E) the extended resistance form on H.…”