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.

拟对称同胚定义了通用Teichmüller空间的一个元素，而对称的同构属于它的小子空间。我们显示了一个实线\（{\ mathbb {R}} \）对称对称同胚h的示例，使得\（h ^ {-1} \）不对称。这意味着\（{\ mathbb {R}} \）的所有对称自同胚性的集合不构成该组合下的组。对于由对称分析的某些概念定义的\（{\ mathbb {R}} \）的强对称自同胚，我们也考虑了相同的问题。这些结果揭示了实线的这种自同胚的集合与单位圆的自同胚的集合的差异。