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2015•Transactions of the American Mathematical SocietyRequires access

Quasihyperbolic metric and Quasisymmetric mappings in metric spaces

Xiaojun Huang, Jinsong Liu

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Abstract

In this paper, we prove that the quasihyperbolic metrics are quasi-invariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of these theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping.

About this research paper

What this paper is about

In this paper, we prove that the quasihyperbolic metrics are quasi-invariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of these theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping.

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OpenAlex reports 29 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

In this paper, we prove that the quasihyperbolic metrics are quasi-invariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of these theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping.

Key concepts: Mathematics, Invariant (physics), Metric space, Metric map, Convex metric space, Injective metric space, Metric (unit), Pure mathematics

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