Conjugating homeomorphisms to uniform homeomorphisms

Authors:
Katsuro Sakai and Raymond Y. Wong

Journal:
Trans. Amer. Math. Soc. **311** (1989), 337-356

MSC:
Primary 58D05; Secondary 57N20, 57S05, 58D15

DOI:
https://doi.org/10.1090/S0002-9947-1989-0974780-0

MathSciNet review:
974780

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Abstract: Let denote the group of homeomorphisms of a metric space onto itself. We say that is conjugate to if for some . In this paper, we study the questions: When is conjugate to which is a uniform homeomorphism or can be extended to a homeomorphism on the metric completion of Typically for a complete metric space , we prove that is conjugate to a uniform homeomorphism if is uniformly approximated by uniform homeomorphisms. In case , we obtain a stronger result showing that every homeomorphism on is, in fact, conjugate to a smooth Lipschitz homeomorphis. For a noncomplete metric space , we provide answers to the existence of under several different settings. Our results are concerned mainly with infinite-dimensional manifolds.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0974780-0

Keywords:
Homeomorphism,
uniform homeomorphism,
conjugation

Article copyright:
© Copyright 1989
American Mathematical Society