Assouad and lower dimensions of graph-directed Bedford-McMullen carpets

TL;DR

This paper computes the Assouad and lower dimensions of graph-directed Bedford-McMullen carpets, revealing their extreme local scaling behaviors and conditions under which these dimensions coincide with Hausdorff and box dimensions.

Contribution

It provides explicit calculations of Assouad and lower dimensions for these carpets and establishes conditions linking different fractal dimensions.

Findings

Assouad and lower dimensions are explicitly calculated.

Conditions are identified for when box and Assouad dimensions coincide.

Hausdorff dimension matches the box dimension under specific conditions.

Abstract

We calculate the Assouad and lower dimensions of graph-directed Bedford-McMullen carpets, which reflect the extreme local scaling laws of the sets, in contrasting with known results on Hausdorff and box dimensions. We also investigate the relationship between distinct dimensions. In particular, we identify an equivalent condition when the box and Assouad dimension coincide, and show that under this condition, the Hausdorff dimension attains the same value.