arXiv subjects

Yoshito Ishiki

Publications and source records attributed to Yoshito Ishiki.

28 records · Page 2Linked to original sources

arXiv2022

Fractal dimensions in the Gromov--Hausdorff space

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.

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arXiv2022

Branching geodesics of the Gromov--Hausdorff distance

In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov--Hausdorff distance. We then construct branching geodesics of the Gromov--Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov--Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.

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arXiv2022

Simultaneous extensions of metrics and ultrametrics of high power

In this paper, generalized metrics mean metrics taking values in general linearly ordered Abelian groups. Using the Hahn fields, we first prove that for every generalized metric space, if the set of the Archimedean equivalence classes of the range group of the metric has an infinite decreasing sequence, then every non-empty closed subset of the space is a uniform retract of the ambient space. Next we construct simultaneous extensions of generalized metrics and ultrametrics. From the existence of extensors of generalized metrics, we characterize the final compactness of generalized metrizable spaces using the completeness of generalized metrics.

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arXiv2022

Continua in the Gromov--Hausdorff space

We first prove that for all compact metrizable spaces, there exists a topological embedding of the compact metrizable space into each of the sets of compact metric spaces which are connected, path-connected, geodesic, or CAT(0), in the Gromov--Hausdorff space with finite prescribed values. As its application, we show that the sets prescribed above are path-connected and their non-empty open subsets have infinite topological dimension. By the same method, we also prove that the set of all proper CAT(0) spaces is path-connected and its non-empty open subsets have infinite topological dimension with respect to the pointed Gromov--Hausdorff distance.

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arXiv2021

Metric trees in the Gromov--Hausdorff space

Using the wedge sum of metric spaces, for all compact metrizable spaces, we construct a topological embedding of the compact metrizable space into the set of all metric trees in the Gromov--Hausdorff space with finite prescribed values. As its application, we show that the set of all metric trees is path-connected and its all non-empty open subsets have infinite topological dimension.

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arXiv2021

On dense subsets in spaces of metrics

In spaces of metrics, we investigate topological distributions of the doubling property, the uniform disconnectedness, and the uniform perfectness, which are the quasi-symmetrically invariant properties appearing in the David--Semmes theorem. We show that the set of all doubling metrics and the set of all uniformly disconnected metrics are dense in spaces of metrics on finite-dimensional and zero-dimensional compact metrizable spaces, respectively. Conversely, this denseness of the sets implies the finite-dimensionality, zero-dimensionality, and the compactness of metrizable spaces. We also determine the topological distribution of the set of all uniformly perfect metrics in the space of metrics on the Cantor set.

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arXiv2021

A characterization of metric subspaces of full Assouad dimension

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.

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arXiv2021

An embedding, an extension, and an interpolation of ultrametrics

The notion of the ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the Arens--Eells isometric embedding theorem of metric spaces, the Hausdorff extension theorem of metrics, the Niemytzki--Tychonoff characterization theorem of the compactness, and the author's interpolation theorem of metrics and theorems on dense subsets of spaces of metrics.

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arXiv2020

On the Assouad dimension and convergence of metric spaces

We introduce the notion of pseudo-cones of metric spaces as a generalization of both of the tangent cones and the asymptotic cones. We prove that the Assouad dimension of a metric space is bounded from below by that of any pseudo-cone of it. We exhibit a example containing all compact metric spaces as pseudo-cones, and examples containing all proper length spaces as tangent cones or asymptotic cones.

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arXiv2019

Quasi-symmetric invariant properties of Cantor metric spaces

For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David-Semmes uniformization theorem states that if a compact metric space satisfies all the three properties, then it is quasi-symmetrically equivalent to the middle-third Cantor set. We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic. In this paper, we conclude that for each of exotic types the class of all the conformal gauges of Cantor metric spaces has continuum cardinality. As a byproduct of our study, we state that there exists a Cantor metric space with prescribed Hausdorff dimension and Assouad dimension.

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