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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 974780
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Abstract: Let $ H(X)$ denote the group of homeomorphisms of a metric space $ X$ onto itself. We say that $ h \in H(X)$ is conjugate to $ g \in H(X)$ if $ {g = fhf^{-1}}$ for some $ f \in H(X)$. In this paper, we study the questions: When is $ h \in H(X)$ conjugate to $ g \in H(X)$ which is a uniform homeomorphism or can be extended to a homeomorphism $ \tilde{g}$ on the metric completion of $ X$ Typically for a complete metric space $ X$, we prove that $ h \in H(X)$ is conjugate to a uniform homeomorphism if $ H$ is uniformly approximated by uniform homeomorphisms. In case $ X = \mathbf{R}$, we obtain a stronger result showing that every homeomorphism on $ \mathbf{R}$ is, in fact, conjugate to a smooth Lipschitz homeomorphis. For a noncomplete metric space $ X$, we provide answers to the existence of $ \tilde{g}$ under several different settings. Our results are concerned mainly with infinite-dimensional manifolds.

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Keywords: Homeomorphism, uniform homeomorphism, conjugation
Article copyright: © Copyright 1989 American Mathematical Society

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