Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T15:00:30.805Z Has data issue: false hasContentIssue false
  • English
  • Français

On Bohr almost periodicity

Published online by Cambridge University Press:  24 October 2008

Paul Milnes
Paul Milnes
Affiliation:
Department of Mathematics, University of Western Ontario, London, Ont. N6A 5B7, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

The first examples of Bohr almost periodic functions that are not almost periodic were given by T. -S. Wu. Later, the present author showed that Bohr almost periodic functions could be distal (and not almost periodic) and even merely minimal. Here it is proved that all Bohr almost periodic functions are minimal. The proof yields an unusual feature about the orbit of a Bohr almost periodic function, one which does not characterize Bohr almost periodic functions, but can be used to show that a Bohr almost periodic function f that is point distal must be distal or, if f is almost automorphic, it must be almost periodic. Some pathologies of Bohr almost periodic functions are discussed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 3 , May 1986 , pp. 489 - 493
Copyright
Copyright © Cambridge Philosophical Society 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Berglund, J. F., Junghenn, H. D. and Milnes, P.. Comp act Right Topological Semigroup and Generalizations of Almost Periodicity. Lecture Notes in Mathematics, 663 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[2]Flor, P.. Rhythmische Abbildungen abelscher Gruppen II. Z. Wahrsch. Verw. Gebiete 7 (1967), 17–28.CrossRefGoogle Scholar
[3]Gottschalk, W. and Hedlund, G.. Topological Dynamics, AMS Colloquium Publication no. 36. American Mathematical Society, 1955).CrossRefGoogle Scholar
[4]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis I (Springer-Verlag, 1963).Google Scholar
[5]Knapp, A. W.. Decomposition theorem for bounded uniformly continuous functions on a group. Amer. J. Math. 88 (1966), 902–914.CrossRefGoogle Scholar
[6]Knapp, A. W.. Functions behaving like almost automorphic functions. Topological Dynamics, an International Symposium (Benjamin, 1966), pp. 299–317.Google Scholar
[7]Milnes, P.. Almost automorphic functions and totally bounded groups. Rocky Mountain J. Math. 7 (1977), 231–250.CrossRefGoogle Scholar
[8]Milnes, P.. Continuity properties of compact right topological groups. Math. Scand. Cambridge Philos. Soc. 86 (1979), 427–435.CrossRefGoogle Scholar
[9]Milnes, P.. Minimal and distal functions on some non-abelian groups. Math. Scand. 49 (1981), 86–94.CrossRefGoogle Scholar
[10]Milnes, P.. The Bohr almost periodic functions do not form a linear space. Math. Z. 188 (1984), 1–2.CrossRefGoogle Scholar
[11]Milnes, P.. Minimal and distal functions on semidirect products of groups. I and II.Google Scholar
[12]Veech, W. A.. Almost automorphic functions on groups. Amer. J. Math. 87 (1965), 719–751.CrossRefGoogle Scholar
[13]Veech, W. A.. Point-distal flows. Amer. J. Math. 92 (1970), 205–242.CrossRefGoogle Scholar
[14]Wu, T.-S.. Left almost periodicity does not imply right almost periodicity. Bull. Amer. Math. Soc. 72 (1966), 314–316.CrossRefGoogle Scholar