Conjugating homeomorphisms to uniform homeomorphisms
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- by Katsuro Sakai and Raymond Y. Wong PDF
- Trans. Amer. Math. Soc. 311 (1989), 337-356 Request permission
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.References
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Additional Information
- © Copyright 1989 American Mathematical Society
- 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