Conjugating homeomorphisms to uniform homeomorphisms
Authors:
Katsuro Sakai and Raymond Y. Wong
Journal:
Trans. Amer. Math. Soc. 311 (1989), 337356
MSC:
Primary 58D05; Secondary 57N20, 57S05, 58D15
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 infinitedimensional manifolds.
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 , Lectures on Hilbert cube manifolds, CBMS Regional Conf. Ser. in Math., no. 28, Amer. Math. Soc., Providence, R.I., 1976. MR 0423357 (54:11336)
 [Cu]
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 [Cu]
 , Boundary sets in the Hilbert cube, Topology Appl. 20 (1985), 201221. MR 804034 (87d:57014)
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 A. H. Kruse and P. W. Liebnitz, An application of a family homotopy extension theorem to ANR spaces, Pacific J. Math. 16 (1966), 331336. MR 0195091 (33:3296)
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 , A manifold localcompactification of a metric combinatorial manifold, Proc. Amer. Math. Soc. 100 (1987), 775780. MR 894453 (88f:57026)
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 L. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271294. MR 0295365 (45:4431)
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 H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), 247262. MR 611763 (82i:57016)
 [To]
 , A correction of two papers concerning Hilbert manifolds, Fund. Math. 125 (1985), 8993. MR 813992 (87m:57017)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198909747800
PII:
S 00029947(1989)09747800
Keywords:
Homeomorphism,
uniform homeomorphism,
conjugation
Article copyright:
© Copyright 1989 American Mathematical Society
