## Locally compact flows on connected manifolds

HTML articles powered by AMS MathViewer

- by Khadija Ben Rejeb
- Conform. Geom. Dyn.
**25**(2021), 253-260 - DOI: https://doi.org/10.1090/ecgd/366
- Published electronically: December 9, 2021
- PDF | Request permission

## Abstract:

In this paper, we completely characterize locally compact flows $G$ of homeomorphisms of connected manifolds $M$ by proving that they are either circle groups or real groups. For $M = \mathbb R^m$, we prove that every recurrent element in $G$ is periodic, and we obtain a generalization of the result of Yang [*Hilbert’s fifth problem and related problems on transformation groups*, American Mathematical Society, Providence, RI, 1976, pp. 142–146.] by proving that there is no nontrivial locally compact flow on $\mathbb R^m$ in which all elements are recurrent.

## References

- Khadija Ben Rejeb,
*Positively equicontinuous flows*, Dyn. Syst.**29**(2014), no. 4, 502–516. MR**3265616**, DOI 10.1080/14689367.2014.947243 - H. Bohr,
*Collected Mathematical works*, Vol II, Danish Math.soc, Univ of Copenhagen (1952). - Khadija Ben Rejeb and Ezzeddine Salhi,
*Characterizations of almost periodic homeomorphisms*, Topology Appl.**158**(2011), no. 16, 2094–2102. MR**2831894**, DOI 10.1016/j.topol.2011.06.008 - J. de Vries,
*Elements of topological dynamics*, Mathematics and its Applications, vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. MR**1249063**, DOI 10.1007/978-94-015-8171-4 - Andreas Dress,
*Newman’s theorems on transformation groups*, Topology**8**(1969), 203–207. MR**238353**, DOI 10.1016/0040-9383(69)90010-X - N. E. Foland,
*The structure of the orbits and their limit sets in continuous flows*, Pacific J. Math.**13**(1963), 563–570. MR**157060**, DOI 10.2140/pjm.1963.13.563 - 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**, DOI 10.1090/coll/036 - V. V. Gorbatsevich, A. L. Onishchik, and E. B. Vinberg,
*Foundations of Lie theory and Lie transformation groups*, Springer-Verlag, Berlin, 1997. Translated from the Russian by A. Kozlowski; Reprint of the 1993 translation [*Lie groups and Lie algebras. I*, Encyclopaedia Math. Sci., 20, Springer, Berlin, 1993; MR1306737 (95f:22001)]. MR**1631937** - Karl H. Hofmann and Sidney A. Morris,
*The Lie theory of connected pro-Lie groups*, EMS Tracts in Mathematics, vol. 2, European Mathematical Society (EMS), Zürich, 2007. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. MR**2337107**, DOI 10.4171/032 - Deane Montgomery and Leo Zippin,
*Topological transformation groups*, Interscience Publishers, New York-London, 1955. MR**0073104** - M.H.A. Newman,
*A theorem on periodic transformations of spaces*, Quart. J. Math,**2**(1931), l–9. - Lex G. Oversteegen and E. D. Tymchatyn,
*Recurrent homeomorphisms on $\textbf {R}^2$ are periodic*, Proc. Amer. Math. Soc.**110**(1990), no. 4, 1083–1088. MR**1037216**, DOI 10.1090/S0002-9939-1990-1037216-3 - John Pardon,
*The Hilbert-Smith conjecture for three-manifolds*, J. Amer. Math. Soc.**26**(2013), no. 3, 879–899. MR**3037790**, DOI 10.1090/S0894-0347-2013-00766-3 - C. T. Yang,
*Hilbert’s fifth problem and related problems on transformation groups*, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R.I., 1976, pp. 142–146. MR**0425999**

## Bibliographic Information

**Khadija Ben Rejeb**- Affiliation: Higher School of Sciences and Technology of Hammam Sousse, 4011 Hammam Sousse, Tunisia
- MR Author ID: 912473
- Email: kbrjeb@yahoo.fr, khadija.benrejeb@essths.u-sousse.tn
- Received by editor(s): April 19, 2020
- Received by editor(s) in revised form: April 1, 2021, and September 5, 2021
- Published electronically: December 9, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**25**(2021), 253-260 - MSC (2020): Primary 37B05, 37B20, 57S05, 57S10
- DOI: https://doi.org/10.1090/ecgd/366
- MathSciNet review: 4349915