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On Quasisymmetric Mappings Between Ultrametric Spaces

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Abstract

In 1980 P. Tukia and J. Väisälä in seminal paper [23] extended a concept of quasisymmetric mapping known from the theory of quasiconformal mappings to the case of general metric spaces. They also found an estimation for the ratio of diameters of two subsets which are images of two bounded subsets of a metric space under a quasisymmetric mapping. We improve this estimation for the case of ultrametric spaces. It was also shown that the image of an ultrametric space under an \(\eta\)-quasisymmetric mapping with \(\eta(1)=1\) is again an ultrametric space. In the case of finite ultrametric spaces it is proved that such mappings are ball-preserving.

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Funding

The research of the first author was partially supported by the National Academy of Sciences of Ukraine, Project 0117U002165 “Development of mathematical models, numerically analytical methods and algorithms for solving modern medico-biological problems”.

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Authors and Affiliations

  1. Function Theory Department, The Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Dobrovolskogo str. 1, 84100, Slovyansk, Ukraine

    Evgeniy Petrov

  2. Complex Analysis and Potential Theory, The Institute of Mathematics of the NAS of Ukraine, Tereschenkivska str. 3, 01024, Kyiv, Ukraine

    Ruslan Salimov

Authors
  1. Evgeniy Petrov
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  2. Ruslan Salimov
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Correspondence to Evgeniy Petrov or Ruslan Salimov.

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Petrov, E., Salimov, R. On Quasisymmetric Mappings Between Ultrametric Spaces. P-Adic Num Ultrametr Anal Appl 13, 231–238 (2021). https://doi.org/10.1134/S2070046621030055

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