On Constructions of Fractal Spaces Using Replacement and the Combinatorial Loewner Property

TL;DR

This paper constructs new fractal spaces using iterated graph systems, disproves Kleiner's conjecture by providing counterexamples, and explores their analysis and conformal properties, expanding understanding of combinatorial Loewner spaces.

Contribution

It introduces a self-similar replacement rule called IGS, provides counterexamples to Kleiner's conjecture, and analyzes their analytical and conformal properties.

Findings

Disproved Kleiner's conjecture with counterexamples.

Introduced iterated graph systems for fractal construction.

Analyzed potentials, conformal dimensions, and analysis on new fractals.

Abstract

The combinatorial Loewner property was introduced by Bourdon and Kleiner as a quasisymmetrically invariant substitute for the Loewner property for general fractals and boundaries of hyperbolic groups. While the Loewner property is somewhat restrictive, the combinatorial Loewner property is very generic -- Bourdon and Kleiner showed that many familiar fractals and group boundaries satisfy it. If $X$ is quasisymmetric to a Loewner space, it has the combinatorial Loewner property. Kleiner conjectured in 2006 that the converse to this holds for self-similar fractals -- the hope being that this would lead to the existence of many exotic Loewner spaces. We disprove this conjecture and give the first examples of spaces which are self-similar, combinatorially Loewner and which are not quasisymmetric to Loewner spaces. In the process we introduce a self-similar replacement rule, called iterated graph systems (IGS), which is inspired by the work of Laakso. This produces a new rich class of fractal spaces, where closed form computations of potentials and their conformal dimensions are possible. These spaces exhibit a rich class of behaviors from analysis on fractals in regards to diffusions, Sobolev spaces, energy measures and conformal dimensions. These behaviors expand on the known examples of Cantor sets, gaskets, Vicsek sets, and the often too difficult carpet-like spaces. Especially the counterexamples to Kleiner's conjecture that arise from this construction are interesting, since they open up the possibility to study the new realm of combinatorially Loewner spaces that are not quasisymmetric to Loewner spaces.