Cartesian products of Sierpiński carpets do not attain their conformal dimension

Abstract

It is a long-standing open question to determine whether the Sierpiński carpet attains its conformal dimension or not. While this problem remains unresolved, we prove that Cartesian products ${\mathbb{S}}^k$, where $\mathbb{S}$ is the Sierpiński carpet and $k\ge 2$, do not attain their conformal dimension. Our approach is based on the Sobolev spaces and energy measures on $\mathbb{S}$ -- constructed by Shimizu, Kigami, and Murugan and Shimizu -- together with a certain singularity result of energy measures from the theory of analysis on fractals. This work formulates a general non-attainment result of conformal dimension for product metric spaces $X^k$ for $k\ge 2$ in terms of self-similarity and energy measures of the factor $X$. It applies, in particular, to the cases where $X$ is the Sierpiński carpet, the Sierpiński gasket, the Menger sponge, and the Laakso diamond.

33 pages. Comments are welcome