Conjugating homeomorphisms to uniform homeomorphisms

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.