A characterization of metric subspaces of full Assouad dimension

TL;DR

This paper introduces tiling spaces for metric spaces, including fractals and Euclidean spaces, and characterizes subspaces with matching Assouad dimension in doubling tiling spaces.

Contribution

It defines tiling spaces for metric spaces and provides a characterization of subspaces with equal Assouad dimension in doubling tiling spaces.

Findings

Tiling spaces include Euclidean and fractal spaces.

Subspaces with full Assouad dimension are characterized in doubling tiling spaces.

Framework applies to various self-similar and fractal spaces.

Abstract

We introduce the notion of tiling spaces for metric spaces. The class of tiling spaces contains the Euclidean spaces, the middle-third Cantor set, and various self-similar spaces appearing in fractal geometry. For doubling tiling spaces, we characterize metric subspaces whose Assouad dimension coincides with that of the whole space.