Thin Loewner Carpets and Their Quasisymmetric Embeddings in S²

Abstract

A carpet is a metric space that is homeomorphic to the standard Sierpiński carpet in ℝ2, or equivalently, in S2. 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′, with 1<Q<Q′<2, we construct infinitely many pairwise quasisymmetrically distinct Q‐Loewner carpets X that admit explicit snowflake embeddings, f:X→S2, for which the image f(X) admits an explicit description and is Q′‐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 S2, so as to yield a circle carpet and also a square carpet. Our Loewner carpets X are constructed via what we call an admissible quotiented inverse system. This mechanism extends the inverse limit construction for PI spaces given in [25], which, however, does not yield carpets. Loewner spaces are a particular subclass of PI spaces. They have strong rigidity properties that do not hold for PI spaces in general. In many cases the construction of the carpets and their snowflake embeddings, f, can also be described in terms of substitution rules. The statement above concerning ()Q,Q′ is already a consequence of these examples. The images of these snowflake embeddings can be de‐snowflaked using a deformation by a strong A∞ weight, which multiplies the metric infinitesimally by a conformal factor of the form ω=dImagef⋅α. Consequently, our examples also yield new examples of strong A∞‐weights for which the associated metrics admit no bi‐Lipschitz embeddings into Banach spaces with the Radon‐Nikodym property such as Lp, for 1<p<normal∞ and ℓ1. © 2021 Wiley Periodicals LLC.