Conductive homogeneity of locally symmetric polygon-based self-similar sets

TL;DR

This work introduces locally symmetric polygon-based self-similar sets defined via $G$-symmetric $J$-gon systems to study conductive homogeneity, a key criterion for constructing Sobolev-type spaces on compact metric spaces and for realizing diffusion processes like Brownian motion on fractal-like sets. The authors establish a structural framework connecting polygon geometry, essential boundary segments, and graph-based conductance to derive sufficient conditions—such as transitive action of $G$ on essential boundary pieces—for $p$-conductive homogeneity when $p>\dim_{AR}(K,d_*)$, with a special emphasis on triangle cases where the result holds unconditionally. Two backbone theorems reduce the problem to combinatorial path constructions on partitions, enabling constructive criteria (including a knight-move criterion) to verify conductivity homogeneity in broad families, including several classical and new polygon-based carpet examples. The work ultimately provides a pathway to define Sobolev-type spaces $\mathcal{W}^p$ and associated Dirichlet forms on these sets, yielding diffusion processes and heat kernel estimates in a non-globally symmetric fractal context. The framework also accommodates cases with isolated boundary points and analyzes how essential boundaries influence the conductive structure and potential regularity of the resulting analytic objects.

Abstract

We provide a rich family of self-similar sets, called locally symmetric polygon-based self-similar sets, as examples of metric spaces having conductive homogeneity, which was introduced as a sufficient condition for the construction of counterparts of "Sobolev spaces" on compact metric spaces. In particular, our results imply the existence of "Brownian motions" on our family of self-similar sets at the same time. Unlike the known examples like the Sierpinski carpet by Barlow-Bass, unconstrained carpet by Cao and Qiu and the Octa-carpet by Andrews, our examples may have no global symmetries, i.e. the group of isometries is trivial.

Figures (19)

• Figure 1: Locally symmetric $J$-gon-based self-similar sets
• Figure 2: $J = 8$, $G = Rot_4$
• Figure 3: Local Symmetry
• Figure 4: Hexagonal cases
• Figure 5: $Q_*^{(5)}$ and $Q^{(6)}_*$

Theorems & Definitions (197)

• definition 1
• Theorem 1.1: = Theorems and
• Theorem 1.2: = Theorem
• definition 2: Regular $J$-gon
• definition 3
• definition 4
• definition 5
• lemma 1
• definition 6
• remark 1