TY - JOUR

T1 - Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion

AU - Wei, Huaying

AU - Matsuzaki, Katsuhiko

N1 - Funding Information:
H. Wei: Research supported by the National Natural Science Foundation of China (Grant No. 11501259).
Funding Information:
K. Matsuzaki: Research supported by Japan Society for the Promotion of Science (KAKENHI 18H01125)
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2021/6

Y1 - 2021/6

N2 - 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.

AB - 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.

KW - Asymptotically conformal

KW - Characteristic topological subgroup

KW - Quasisymmetric

KW - Strongly symmetric

KW - Symmetric homeomorphism

KW - VMO

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U2 - 10.1007/s13324-021-00510-7

DO - 10.1007/s13324-021-00510-7

M3 - Article

AN - SCOPUS:85102569122

VL - 11

JO - Analysis and Mathematical Physics

JF - Analysis and Mathematical Physics

SN - 1664-2368

IS - 2

M1 - 79

ER -