When the conformal dimension of a self-affine sponge of Lalley-Gatzouras type is zero

TL;DR

This work determines when the conformal dimension $\dim_C K$ of a diagonal self-affine sponge $K$ of Lalley-Gatzouras type is zero. By developing a tree-based IFS description and equivalent characterizations of uniform disconnectedness, the authors prove that $\dim_C K=0$ iff $K$ is uniformly disconnected; otherwise $\dim_C K\ge 1$. The proof hinges on (i) a precise equivalence between uniform disconnectedness and fiber IFS properties (notably that no fiber IFS attractor equals $[0,1]$), and (ii) constructing a Lipschitz model $\bar{K}$ that contains a Cantor-type subset $E$ with conformal dimension 1, invoking Hakobyan’s results to conclude $\dim_C K\ge 1$ when not uniformly disconnected. This yields a complete dichotomy for the conformal dimension of Lalley-Gatzouras sponges under quasisymmetric maps and clarifies how fiber structure governs $\dim_C$. The results extend the understanding of conformal dimension beyond self-similar and certain self-affine carpets to a broad class of Lalley-Gatzouras sponges, with potential implications for rigidity and geometric structure under quasisymmetric equivalence.

Abstract

It is well known that if a metric space is uniformly disconnected, then its conformal dimension is zero. First, we characterize when a self-affine sponge of Lalley-Gatzouras type is uniformly disconnected. Thanks to this characterization, we show that a self-affine sponge of Lalley-Gatzouras type has conformal dimension zero if and only if it is uniformly disconnected.

Figures (7)

• Figure 1: IFS for Bedford-McMullen carpet (left) and self-affine carpet of Lalley-Gatzouras type (right). See Example \ref{['Exam:FiberIFS']}.
• Figure 2: Labeled tree of the IFS $\Phi$ on the right side of Figure \ref{['Carpet:Lalley']}. The fiber IFS related to $\frac{x}{3}$ is $\{\phi_1,\phi_2\}$, and the fiber IFS related to $\frac{x+1}{2}$ is $\{\phi_3, \phi_4,\phi_5\}$.
• Figure 3: An example of self-affine carpet of Lalley-Gatzouras type satisfying the conditions in Theorem \ref{['thm:simple-IFS']}.
• Figure 4: The red dashed lines show the biggest gap bigger than $\delta$.
• Figure 5: $\beta_{p+1} \geq \alpha_{p+1} + 2$. The green intervals are elements in ${\mathcal{B}}([u,v])$.

Theorems & Definitions (50)

• Definition 1.1: Uniformly disconnected Heinonen01
• Lemma 1.2: Heinonen01
• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Definition 1.4
• Theorem 1.3
• Remark 1.5
• Theorem 1.4
• Theorem 1.5