Conductive homogeneity of locally symmetric polygon-based self-similar sets
Metric Geometry
2025-05-30 v1
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.
Cite
@article{arxiv.2505.23057,
title = {Conductive homogeneity of locally symmetric polygon-based self-similar sets},
author = {Jun Kigami and Yuka Ota},
journal= {arXiv preprint arXiv:2505.23057},
year = {2025}
}