Cartesian products of Sierpi\'nski carpets do not attain their conformal dimension

TL;DR

This paper proves that Cartesian products of the Sierpiński carpet with itself do not reach their conformal dimension, using Sobolev spaces and energy measures on fractals, extending to other self-similar spaces.

Contribution

It establishes a general non-attainment result of conformal dimension for product spaces based on self-similarity and energy measures, applicable to various fractals.

Findings

Cartesian products of the Sierpiński carpet do not attain their conformal dimension.

The approach uses Sobolev spaces and energy measures on fractals.

The result applies to multiple self-similar fractal spaces.

Abstract

It is a long-standing open question to determine whether the Sierpi\'nski 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 Sierpi\'nski carpet and $k \geq 2$, 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 $k \geq 2$ in terms of self-similarity and energy measures of the factor $X$. It applies, in particular, to the cases where $X$ is the Sierpi\'nski carpet, the Sierpi\'nski gasket, the Menger sponge, and the Laakso diamond.