Assouad type spectra for some self-affine sponges

Abstract

In this paper, we compute the Assouad and lower spectra of Bedford–McMullen sponges in ${\mathbb{R}}^3$ explicitly. According to the formulae established, we discover that the spectra are not determined by the ratio set, and the box, lower and Assouad dimensions of the sponge anymore, which is unlike the situation in a planar carpet. As a by-product, we construct two Bedford–McMullen sponges on the same grid, both of which have non-uniform fibres. Particularly, they share the same box, lower and Assouad dimensions. However, their Assouad type spectra are different and therefore, they are not bi-Lipschitz equivalent. For Bedford–McMullen sponges in higher dimensions, we also determine the dimension spectra when $\theta$ is smaller than the minimal ratio or it is bigger than the maximal ratio.