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

TL;DR

This paper introduces a new family of self-similar sets based on locally symmetric polygons, demonstrating their conductive homogeneity and the existence of Brownian motion on these sets, even without global symmetries.

Contribution

It constructs a broad class of self-similar sets with conductive homogeneity, expanding examples beyond those with global symmetries and enabling analysis of Brownian motion on them.

Findings

Existence of Brownian motion on the new self-similar sets.

Examples include sets with trivial isometry groups.

These sets exhibit conductive homogeneity without global symmetries.

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.