The generalized symmetric Teichmüller spaces T_X^* are defined for finite subsets X of the unit circle and are shown to be complex Banach manifolds via the Bers embedding.
Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion
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abstract
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 deal with strongly symmetric self-homeomorphisms of $\mathbb{R}$ along the same line. 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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Teichm\"uller spaces of generalized symmetric homeomorphisms
The generalized symmetric Teichmüller spaces T_X^* are defined for finite subsets X of the unit circle and are shown to be complex Banach manifolds via the Bers embedding.