Assouad type dimensions for certain sponges with a weak coordinate ordering condition

TL;DR

This paper calculates the Assouad type dimensions for Bedford-McMullen and Lalley-Gatzouras self-affine sponges with weak coordinate ordering, expanding understanding of their geometric complexity.

Contribution

It provides the first explicit formulas for Assouad type dimensions of these sponges under weak coordinate ordering conditions.

Findings

Derived formulas for Assouad dimensions of weak-coordinate Bedford-McMullen sponges.

Extended analysis to Lalley-Gatzouras sponges with similar conditions.

Discussed subtle geometric details affecting dimension calculations.

Abstract

Recently self-affine sponges have been shown to be interesting counter-examples to several previously open problems. One class of recently studied sponges are Bedford-McMullen sponges with a weak coordinate ordering condition, that is, sponges with several coordinates having the same contraction ratio. The Assouad type dimensions of such sets cannot be calculated using the same formula as the regular Bedford-McMullen sponges. We calculate the Assouad type dimensions for such sponges and the more general Lalley-Gatzouras sponges with a weak coordinate ordering condition, discussing some of their more subtle details along the way.