Conformal dimension and its attainment on self-similar Laakso-type fractal spaces

TL;DR

This work resolves the attainment problem for the Ahlfors regular conformal dimension on a broad class of symmetric Laakso-type fractal spaces constructed as limits of iterated graph systems. Attainment is characterized by a simple, verifiable condition on removable edges in the generator: no removable edges implies attainment with an explicit metric $d_{\rho}$ built from a symmetric $\Theta^{(1)}$-admissible density, while removable edges force porosity and non-attainment. The conformal dimension is identified as the unique $Q_*$ solving ${\rm Mod}_{Q_*}(\Theta^{(1)},G_1)=1$, with modulus scaling multiplicatively along levels $${\rm Mod}_p(\Theta^{(n)},G_n)={\rm Mod}_p(\Theta^{(1)},G_1)^n.$$ The results connect attainment to Loewner theory and Kleiner's conjecture, showing that spaces without removable edges are quasisymmetric to Loewner spaces and that the associated Sobolev-energy framework aligns with attainment through a doubling energy-dominant measure.

Abstract

A general construction of Laakso-type fractal spaces was recently introduced by the first two authors. In this paper, we establish a simple condition characterizing when the Ahlfors regular conformal dimension of a symmetric Laakso-type fractal space is attained. The attaining metrics are constructed explicitly. This gives new examples of attainment and clarifies the possible obstructions.

Figures (4)

• Figure 1.0: First iteration of the procedure that produces a sequence of graphs converging to a limit space known as the Laakso diamond.
• Figure 1.0: The Laakso diamond.
• Figure 1.1: The generator $G_1$ of a symmetric Laakso-type fractal space with gluing sets $I_-=\{v^-,w^-\}$ and $I_+=\{v^+,w^+\}$. $G_2$ is produced by replacing each edge of $G_1$ with a copy of itself and gluing the copies along the gluing sets at the junction $u$. The symmetry exchanges $v^+$ with $v^-$ and $w^+$ with $w^-$.
• Figure 1.4: Two generators for symmetric Laakso-type spaces. The one on the left includes a vertical edge positioned in the middle of the graph. This edge is invariant under the symmetry, and thus must be removable. This example was the first one to be found to contradict Kleiner's conjecture, see anttila2024constructions. The graph on the right does not have a removable edge, thereby giving an example where attainment occurs and the conjecture holds by Theorem \ref{['Main Theorem: Characterization of Attainment']} and Corollary \ref{['cor:Loewner']}. This graph is obtained by taking two parallel edges connecting two vertices in the middle, and adjoining three edges to each side in order to connect the boundary.

Theorems & Definitions (89)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.3
• Definition 2.4
• Lemma 2.5: He
• Corollary 2.6
• proof