Almost periodic homeomorphisms and $p$-adic transformation groups on compact $3$-manifolds

Author:
Joo S. Lee

Journal:
Proc. Amer. Math. Soc. **121** (1994), 267-273

MSC:
Primary 57M60; Secondary 57S20, 57S25

DOI:
https://doi.org/10.1090/S0002-9939-1994-1233977-2

MathSciNet review:
1233977

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 Hilbert-Smith conjecture on compact 3-manifolds ${M^3}$: If *h* is almost periodic on ${M^3}$, with *h* = identity on $\partial {M^3}$, then *h* = identity on ${M^3}$.

- Beverly L. Brechner,
*Almost periodic homeomorphisms of $E^{2}$ are periodic*, Pacific J. Math.**59**(1975), no. 2, 367–374. MR**388361**
---, - Glen E. Bredon,
*Introduction to compact transformation groups*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR**0413144** - Andreas Dress,
*Newman’s theorems on transformation groups*, Topology**8**(1969), 203–207. MR**238353**, DOI https://doi.org/10.1016/0040-9383%2869%2990010-X - N. E. Foland,
*A characterization of the almost periodic homeomorphisms on the closed $2$-cell*, Proc. Amer. Math. Soc.**16**(1965), 1031–1034. MR**180958**, DOI https://doi.org/10.1090/S0002-9939-1965-0180958-8 - Walter Helbig Gottschalk and Gustav Arnold Hedlund,
*Topological dynamics*, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955. MR**0074810** - W. H. Gottschalk,
*Minimal sets: an introduction to topological dynamics*, Bull. Amer. Math. Soc.**64**(1958), 336–351. MR**100048**, DOI https://doi.org/10.1090/S0002-9904-1958-10223-2
H. T. Ku, M. C. Ku, and L. M. Mann, - Frank Raymond,
*Cohomological and dimension theoretical properties of orbit spaces of $p$-adic actions*, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 354–365. MR**0260925** - P. A. Smith,
*Transformations of finite period. II*, Ann. of Math. (2)**40**(1939), 690–711. MR**177**, DOI https://doi.org/10.2307/1968950 - P. A. Smith,
*Transformations of finite period. III. Newman’s theorem*, Ann. of Math. (2)**42**(1941), 446–458. MR**4128**, DOI https://doi.org/10.2307/1968910 - P. A. Smith,
*Periodic and nearly periodic transformations*, Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., 1941, pp. 159–190. MR**0005302** - E. R. Van Kampen,
*The Topological Transformations of a Simple Closed Curve Into Itself*, Amer. J. Math.**57**(1935), no. 1, 142–152. MR**1507062**, DOI https://doi.org/10.2307/2372026 - Chung-Tao Yang,
*$p$-adic transformation groups*, Michigan Math. J.**7**(1960), 201–218. MR**120310**

*Prime ends and group actions*, preprint.

*Newman’s theorem and the Hilbert-Smith conjecture*, Group Actions on Manifolds, Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 489-491. L. F. McAuley,

*A p-adic group cannot act effectively as a transformation group on a n-manifold*(

*Hilbert Fifth Problem*), preprint. M. H. A. Newman,

*A theorem on periodic transformations of spaces*, Quart. J. Math

**2**(1931), 1-9.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57M60,
57S20,
57S25

Retrieve articles in all journals with MSC: 57M60, 57S20, 57S25

Additional Information

Keywords:
Manifold,
almost periodic homeomorphism,
the Hilbert-Smith conjecture

Article copyright:
© Copyright 1994
American Mathematical Society