Almost periodic homeomorphisms and adic transformation groups on compact manifolds
Author:
Joo S. Lee
Journal:
Proc. Amer. Math. Soc. 121 (1994), 267273
MSC:
Primary 57M60; Secondary 57S20, 57S25
MathSciNet review:
1233977
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Abstract: In this paper we prove that regularly almost periodic is equivalent to nearly periodic for homeomorphisms on compact metric spaces and give an example to show that the above is false without the compactness assumption. We also prove that the following statement is equivalent to the HilbertSmith conjecture on compact 3manifolds : If h is almost periodic on , with h = identity on , then h = identity on .
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 B. Brechner, Almost periodic homeomorphisms of are periodic, Pacific J. Math. 59 (1975), 367374. MR 0388361 (52:9198)
 [B2]
 , Prime ends and group actions, preprint.
 [Br]
 G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York and London, 1972. MR 0413144 (54:1265)
 [D]
 A. Dress, Newman's theorem on transformation groups, Topology 8 (1969), 203207. MR 0238353 (38:6629)
 [F]
 N. E. Foland, A characterization of the almost periodic homeomorphisms on the closed 2cell, Proc. Amer. Math. Soc. 16 (1965), 10311034. MR 0180958 (31:5188)
 [G1]
 W. H. Gottschalk, Topological dynamics, Amer. Math. Soc. Colloq. Publ. vol. 36, Amer. Math. Soc., Providence, RI, 1955. MR 0074810 (17:650e)
 [G2]
 , Minimal sets; An introduction topological dynamics, Bull. Amer. Math. Soc. 64 (1958), 336351. MR 0100048 (20:6484)
 [K]
 H. T. Ku, M. C. Ku, and L. M. Mann, Newman's theorem and the HilbertSmith conjecture, Group Actions on Manifolds, Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 489491.
 [M]
 L. F. McAuley, A padic group cannot act effectively as a transformation group on a nmanifold (Hilbert Fifth Problem), preprint.
 [N]
 M. H. A. Newman, A theorem on periodic transformations of spaces, Quart. J. Math 2 (1931), 19.
 [R]
 F. Raymond, Cohomological and dimension theoretical properties of orbit spaces of padic actions, Proceedings of Conf. Transformation Groups (New Orleans), SpringerVerlag, Berlin and New York, 1968, pp. 354365. MR 0260925 (41:5545)
 [S1]
 P. A. Smith, Transformations of finite period: II, Ann. of Math. 40 (1939), 690711. MR 0000177 (1:30c)
 [S2]
 , Transformations of finite period: III, Ann. of Math. 42 (1941), 446458. MR 0004128 (2:324c)
 [S3]
 , Periodic and nearly periodic transformations, Lectures in Topology, University of Michigan Press, Ann Arbor, MI, 1941, pp. 159190. MR 0005302 (3:133c)
 [vK]
 E. van Kampen, The topological transformation of a simple closed curve into itself, Amer. J. Math. 57 (1935), 142152. MR 1507062
 [Y]
 C. T. Yang, padic transformation groups, Michigan Math. J. 7 (1960), 201218. MR 0120310 (22:11065)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199412339772
PII:
S 00029939(1994)12339772
Keywords:
Manifold,
almost periodic homeomorphism,
the HilbertSmith conjecture
Article copyright:
© Copyright 1994
American Mathematical Society
