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

Jun Kigami; Yuka Ota

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arXiv · 2505.23057

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

Jun Kigami·Yuka Ota

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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.

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Jun Kigami, Yuka Ota. 2025-05-29. Conductive homogeneity of locally symmetric polygon-based self-similar sets. https://arxiv.org/abs/2505.23057

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