Fractal dimensions in the Gromov--Hausdorff space

TL;DR

This paper constructs specific ultrametric spaces with prescribed fractal dimensions and demonstrates the topological properties of sets of compact metric and ultrametric spaces within the Gromov--Hausdorff space, revealing their path-connectedness and infinite dimension.

Contribution

It introduces a method to realize any four given fractal dimensions in ultrametric spaces and proves the path-connectedness and infinite topological dimension of certain sets in the Gromov--Hausdorff space.

Findings

Existence of ultrametric spaces with prescribed fractal dimensions.

Path-connectedness of sets of compact metric and ultrametric spaces.

Infinite topological dimension of these sets.

Abstract

In this paper, we first show that for all four non-negative real numbers, there exists a Cantor ultrametric space whose Hausdorff dimension, packing dimension, upper box dimension, and Assouad dimension are equal to given four numbers, respectively. Next, by constructing topological embeddings of an arbitrary compact metrizable space into the Gromov--Hausdorff space using a direct sum of metrics spaces, we prove that the set of all compact metric spaces possessing prescribed topological dimension, and four dimensions explained above, and the set of all compact ultrametric spaces are path-connected and have infinite topological dimension. This observation on ultrametrics provides another proof of Qiu's theorem stating that the ratio of the Archimedean and non-Archimedean Gromov--Hausdorff distances is unbounded.