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Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion

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Abstract

A quasisymmetric homeomorphism defines an element of the universal Teichmüller space and a symmetric one belongs to its little subspace. We show an example of a symmetric homeomorphism h of the real line \({\mathbb {R}}\) onto itself such that \(h^{-1}\) is not symmetric. This implies that the set of all symmetric self-homeomorphisms of \({\mathbb {R}}\) does not constitute a group under the composition. We also consider the same problem for a strongly symmetric self-homeomorphism of \({\mathbb {R}}\) which is defined by a certain concept of harmonic analysis. These results reveal the difference of the sets of such self-homeomorphisms of the real line from those of the unit circle.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, PR China

    Huaying Wei

  2. Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo, 169-8050, Japan

    Katsuhiko Matsuzaki

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  1. Huaying Wei
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  2. Katsuhiko Matsuzaki
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Correspondence to Katsuhiko Matsuzaki.

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H. Wei: Research supported by the National Natural Science Foundation of China (Grant No. 11501259).

K. Matsuzaki: Research supported by Japan Society for the Promotion of Science (KAKENHI 18H01125)

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Wei, H., Matsuzaki, K. Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion. Anal.Math.Phys. 11, 79 (2021). https://doi.org/10.1007/s13324-021-00510-7

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  • DOI: https://doi.org/10.1007/s13324-021-00510-7

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