Fractal dimensions in the Gromov–Hausdorff space
Tom 71 / 2023
Bulletin Polish Acad. Sci. Math. 71 (2023), 147-168
MSC: Primary 53C23; Secondary 51F99
DOI: 10.4064/ba221212-16-11
Opublikowany online: 11 December 2023
Streszczenie
We first show that for any 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 the given four numbers, respectively. Next, using a direct sum of metric spaces, we construct topological embeddings of an arbitrary compact metrizable space into the two subsets of the Gromov–Hausdorff space: the set of all compact metric spaces possessing prescribed topological dimension and the aforementioned four dimensions, and the set of all compact ultrametric spaces.
