
About some properties of similarly homogeneous $\mathbb {R}$trees
A. I. Bulygin^{} ^{} Northern (Arctic) Federal University named after M. V. Lomonosov, 17 Severnaya Dvina Emb., Arkhangelsk 163002, Russia
Abstract:
In this paper we consider
the properties of locally complete similarly homogeneous
inhomogeneous $\mathbb{R}$trees. The geodesic space is called $\mathbb{R}$tree
if any two points may be connected by the unique arc. The general problem of
A. D. Alexandrov on the characterization of metric spaces is considered. The distance
one preserving mappings are constructed for some classes of $\mathbb{R}$trees. To do
this, we use the construction with the help of which a new special metric is introduced
on an arbitrary metric space. In terms of this new metric, a criterion is formulated that
is necessary for a so that a distance one preserving mapping to be isometric. In this
case, the characterization by A. D. Alexandrov is not fulfilled.
Moreover, the boundary of a strictly vertical $\mathbb{R}$tree is also studied.
It is proved that any horosphere in a strictly vertical $\mathbb{R}$tree is an
ultrametric space. If the branch number of a strictly vertical $\mathbb{R}$tree is not
greater than the continuum, then the cardinality of any sphere and any horosphere in the
$\mathbb{R}$tree equals the continuum, and if the branch number
of $\mathbb{R}$tree is larger than the continuum, then the cardinality
of any sphere or horosphere equals the
number of branches.
Key words:
similarly homogeneous space,
strictly vertical $\mathbb{R}$tree, metric, horosphere.
DOI:
https://doi.org/10.23671/VNC.2020.1.57537
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UDC:
515.124
MSC: 54E35, 54E45 Received: 15.07.2019
Citation:
A. I. Bulygin, “About some properties of similarly homogeneous $\mathbb {R}$trees”, Vladikavkaz. Mat. Zh., 22:1 (2020), 32–42
Citation in format AMSBIB
\Bibitem{Bul20}
\by A.~I.~Bulygin
\paper About some properties of similarly homogeneous $\mathbb {R}$trees
\jour Vladikavkaz. Mat. Zh.
\yr 2020
\vol 22
\issue 1
\pages 3242
\mathnet{http://mi.mathnet.ru/vmj712}
\crossref{https://doi.org/10.23671/VNC.2020.1.57537}
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