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

TL;DR

This paper characterizes when Lalley-Gatzouras self-affine sponges are uniformly disconnected and shows that their conformal dimension is zero precisely in those cases, linking geometric structure to conformal properties.

Contribution

It provides a complete characterization of uniform disconnectedness for Lalley-Gatzouras sponges and establishes the equivalence with having conformal dimension zero.

Findings

A self-affine sponge of Lalley-Gatzouras type is conformal dimension zero if and only if it is uniformly disconnected.

The paper offers a geometric criterion for uniform disconnectedness in these sponges.

It advances understanding of the relationship between fractal geometry and conformal invariants.

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.