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On metric geometry of convergences and topological distributions of metric structures

伊敷, 喜斗 筑波大学 DOI:10.15068/0002000938

2021.08.03

概要

In this doctoral thesis, we investigate metric geometry from the viewpoints of convergences and topological distributions of metric structure. This thesis mainly consists of two parts. The fist part is a study on the Assouad dimension and limit spaces of subspaces of metric spaces. We introduce pseudo-cones of metric spaces as generalizations of tangent and asymptotic cones of metric spaces, and provide lower estimations of the Assouad dimensions of metric spaces by the dimensions of pseudo-cones. This is a generalization of the Mackay–Tyson estimation of the Assouad dimension by tangent cones. As another application of pseudo-cones, we study subsets of full Assouad dimension of metric spaces, and we introduce the notion of a tiling space. A tiling space is a pair of a metric space and a family of subsets called tiles of the metric spaces. The class of tiling spaces contains the Euclidean spaces, the p-adic numbers, the Sierpin´ski gasket, and various self-similar spaces appearing in fractal geometry. As our result, for a doubling tiling space, we characterize a subspace possessing the same Assouad dimension as that of the whole space in terms of pseudo-cones and tiles. Since the Euclidean spaces are tiling spaces, this result can be considered as a generalization of the Fraser–Yu characterization of a subspace of the Euclidean space of full Assouad dimension.

The second part is a study on topological distributions of sets of “singular” metrics in spaces of metrics. We first prove an interpolation theorem of metrics adapted for investigating topologies of spaces of metrics. We introduce the notion of the transmissible property, which unifies geometric properties determined by finite subsets of metric spaces. As an application of our interpolation theorem, we prove that the sets of all metrics not satisfying transmissible properties are dense and represented as an intersection of countable open subsets of spaces of metrics. We also prove analogues of these results for ultrametric spaces. It is often expected to prove ultrametric analogues of statements on ordinary metrics. As realizations of this expectation, we first prove an isometric embedding theorem stating that every ultrametric space can be isometrically embedded into an ultra- normed module over an integral domain, which is an analogue of the Arens–Eells isometric embedding theorem. Due to this embedding theorem, as an analogue of the Hausdorff metric extension theorem, we can prove a theorem on extending an ultrametric on a closed subset to an ultrametric on the whole space, while referring to the Torun´czyk’s proof of the Hausdorff metric extension theorem by the Arens–Eells isometric embedding theorem. By our extension theorem on ultrametrics, we establish an interpolation theorem on ultrametrics, and theorems on topological distributions on spaces of ultrametrics.

This doctoral thesis is written as a comprehensive paper including the contents of the author’s papers [60], [56], [59], and [58]. Most of the results stated in this doctoral thesis have appeared in [60], [56], [59], and [58]. The author has added some auxiliary explanations and statements for the sake of comprehension of readers.

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