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 R onto itself such that h- 1 is not symmetric. This implies that the set of all symmetric self-homeomorphisms of R does not constitute a group under the composition. We also consider the same problem for a strongly symmetric self-homeomorphism of 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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