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Modulus Characterizations of Bilipschitz Mappings

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Abstract

In this paper, we establish six necessary and sufficient conditions for a homeomorphism of \(\mathbb {R}^n\) onto itself to be strongly quasisymmetric. These conditions are quantitative in terms of conformal moduli of disjoint continua as well as the geometric modulus, which was recently introduced by Tukia and Väisälä. Note that all of them are equivalent to bilipschitz continuity with parameters depending also on two fixed points. As an application, we obtain several quantitative characterizations for a homeomorphism of the Riemann sphere \(\overline{\mathbb {R}}^n\) onto itself to be strongly quasimöbius.

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Acknowledgements

The authors wish to express their sincere gratitude to the anonymous referee, whose extremely careful reading of the manuscript led to many clarifications and improvements in the text.

Funding

Qingshan Zhou was supported by Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515012441). Antti Rasila was supported by NSF of China (No. 11971124), and NSFs of Guangdong province (Nos. 2021A1515010326, 2024A1515010467). Yuehui He was supported by Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515111136).

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Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, 528000, Guangdong, People’s Republic of China

    Qingshan Zhou & Yuehui He

  2. MOE–LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Zhiqiang Yang

  3. Department of Mathematics with Computer Science, Guangdong Technion, 241 Daxue Road, Shantou, 515063, Guangdong, People’s Republic of China

    Antti Rasila

  4. Department of Mathematics, Technion – Israel Institute of Technology, Haifa, 32000, Israel

    Antti Rasila

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  1. Qingshan Zhou
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  2. Zhiqiang Yang
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Correspondence to Yuehui He.

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Zhou, Q., Yang, Z., Rasila, A. et al. Modulus Characterizations of Bilipschitz Mappings. J Geom Anal 34, 155 (2024). https://doi.org/10.1007/s12220-024-01635-4

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