Smooth Lipschitz Research Articles

Conjugating homeomorphisms to uniform homeomorphisms

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