Thin Loewner carpets and their quasisymmetric embeddings in $S^2$

Abstract

A carpet is a metric space which is homeomorphic to the standard Sierpiński carpet in ${\mathbb{R}}^2$, or equivalently, in $S^2$. A carpet is called thin if its Hausdorff dimension is $<2$. A metric space is called Q-Loewner if its $Q$-dimensional Hausdorff measure is Q-Ahlfors regular and if it satisfies a $(1,Q)$-Poincaré inequality. As we will show, $Q$-Loewner planar metric spaces are always carpets, and admit quasisymmetric embeddings into the plane. In this paper, for every pair $(Q,Q^{\prime })$, with $1<Q<Q^{\prime }<2$ we construct infinitely many pairwise quasi-symmetrically distinct $Q$-Loewner carpets $X$ which admit explicit snowflake embeddings, $f:X\to S^2$, for which the image, $f(X)$, admits an explicit description and is $Q^{\prime }$-Ahlfors regular. In particular, these $f$ are quasisymmetric embeddings. By a result of Tyson, the Hausdorff dimension of a Loewner space cannot be lowered by a quasisymmetric homeomorphism. By definition, this means that the carpets $X$ and $f(X)$ realize their conformal dimension. Each of images $f(X)$ can be further uniformized via post composition with a quasisymmetric homeomorphism of $S^2$, so as to yield a circle carpet and also a square carpet. Our Loewner carpets $X$ are constructed via what we call an admissable quotiented inverse system. This mechanism extends the inverse limit construction for PI spaces given in \cite{cheegerkleinerinverse}, which however, does not yield carpets. Loewner spaces are a particular subclass of PI spaces. They have strong rigidity properties which which do not hold for PI spaces in general.

60 Pages, 9 figures, comments welcome. Edited with a slightly stronger result and correcting typos. Acknowledgements and relation to prior work by Bruce Kleiner and Mario Bonk clarified and expanded

References in corpus (2)

• Potential theory on Sierpinski carpets with applications to uniformization
• Quasisymmetric Embeddability of Weak Tangents