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On quasisymmetric mappings between ultrametric spaces

Evgeniy Petrov, Ruslan Salimov

TL;DR

This paper analyzes how $η$-quasisymmetric mappings act on ultrametric spaces, refining diameter-distortion estimates in the ultrametric setting and establishing preservation of ultrametricity under maps with $η(1)=1$. It further shows that surjective, finite, $η$-quasisymmetric embeddings are ball-preserving, yielding isomorphisms of the representing trees that encode ultrametric structure. The results connect analytic mapping properties to the combinatorial representation of ultrametric spaces via rooted trees, highlighting a rigidity of finite ultrametric spaces under quasisymmetric deformations with implications for their ball hierarchies. These insights advance the understanding of how quasisymmetric mappings preserve or reveal the hierarchical organization intrinsic to ultrametric spaces.

Abstract

In 1980 P. Tukia and J. Väisälä in seminal paper [P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn., Ser. A I, Math. 5, 97--114 (1980)] 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 $η$-quasisymmetric mapping with $η(1)=1$ is again an ultrametric space. In the case of finite ultrametric spaces it is proved that such mappings are ball-preserving.

On quasisymmetric mappings between ultrametric spaces

TL;DR

This paper analyzes how -quasisymmetric mappings act on ultrametric spaces, refining diameter-distortion estimates in the ultrametric setting and establishing preservation of ultrametricity under maps with . It further shows that surjective, finite, -quasisymmetric embeddings are ball-preserving, yielding isomorphisms of the representing trees that encode ultrametric structure. The results connect analytic mapping properties to the combinatorial representation of ultrametric spaces via rooted trees, highlighting a rigidity of finite ultrametric spaces under quasisymmetric deformations with implications for their ball hierarchies. These insights advance the understanding of how quasisymmetric mappings preserve or reveal the hierarchical organization intrinsic to ultrametric spaces.

Abstract

In 1980 P. Tukia and J. Väisälä in seminal paper [P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn., Ser. A I, Math. 5, 97--114 (1980)] 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 -quasisymmetric mapping with is again an ultrametric space. In the case of finite ultrametric spaces it is proved that such mappings are ball-preserving.
Paper Structure (3 sections, 14 theorems, 41 equations)

This paper contains 3 sections, 14 theorems, 41 equations.

Table of Contents

  1. Introduction.
  2. Distortion of the ratio of diameters
  3. Ultrametric-preserving $\eta$-quasisymmetric embeddings.

Key Result

Proposition 1.2

Let $X, Y$ be metric spaces and let $f$ be an $\eta$-quasisymmetric embedding. Let $A\subset B\subset X$ with $\mathop{\mathrm{diam}}\nolimits(A)>0$, $\mathop{\mathrm{diam}}\nolimits(B)<\infty$. Then $\mathop{\mathrm{diam}}\nolimits(fB)<\infty$ and

Theorems & Definitions (24)

  • Definition 1.1: TV80
  • Proposition 1.2
  • Definition 1.3: DDP(P-adic)
  • Theorem 1.4: DDP(P-adic)
  • Lemma 1.5
  • Proposition 1.6
  • Proposition 2.1: TV80
  • proof
  • Theorem 2.2
  • proof
  • ...and 14 more