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

Abstract

A carpet is a metric space which is homeomorphic to the standard Sierpinski carpet in 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 ), with 1 < Q < Q < 2 we construct infinitely many pairwise quasi-symmetrically distinct Q-Loewner carpets X which admit explicit snowflake embeddings, f : X → S 2 , 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 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 [25], 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.In many cases the construction of the carpets and their snowflake embeddings, f , can also be described in terms of replacement 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 = d(Image(f ), •) 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 < ∞ and 1.

Uniformizing Gromov hyperbolic spaces and Busemann functions

“…Uniform domains were independently introduced by John in [39] and by Martio and Sarvas in [53]. The importance of these domains in Euclidean spaces and metric spaces arises from their connections to various results in geometric function theory, such as the extendability of functions in Sobolve spaces [40], the characterizations of Gromov hyperbolic domains [7,58], Teichmüller's displacement problem [51,64], the quasisymmetric embedding of planar Loewner carpets in S 2 [16], and even the constructions of metric spaces with empty interior but supporting Poincaé inequalities [20], and so on.…”