Whether $p$-conductive homogeneity holds depends on $p$

Abstract

We introduce two fractals, in Euclidean spaces of dimension two and three, respectively, such that the $2$-conductive homogeneity holds but there is some $\epsilon \in (0,1)$ so that the $p$-conductive homogeneity fails for every $p\in (1,1+\epsilon )$. In addition, these two fractals have Ahlfors regular conformal dimension within the interval $(1,2)$ and $(2,3)$, respectively.