Conductive homogeneity of polygon-based self similar sets

Last year, I gave a talk in the probability seminar on how to construct an analogue of “Sobolev spaces” on compact metric spaces. The idea was to approximate the original space by a sequence of graphs, consider discrete \$p\$-energies on those graphs, and take a scaling limit of them.
In this talk, what I will present is new examples of spaces where this strategy works.
More precisely,  I will consider polygon-based locally symmetric self-similar sets. One of the results is that the above strategy works for any triangle-based cases.