A characterization of distal and point-distal minimal transformation groups
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- by Joseph F. Kent PDF
- Proc. Amer. Math. Soc. 32 (1972), 304-308 Request permission
Abstract:
If G is a locally compact topological group, let $BC(G)$ denote the set of real-valued, bounded, uniformly continuous functions on G with the compact-open topology. Using the fact that the distal (weakly distal) functions are the elements of $BC(G)$ whose orbit closures are compact distal (point-distal) minimal sets, we can characterize compact distal and point-distal minimal transformation groups. Let $(X,G,\phi )$ be a right transformation group where X is compact Hausdorff and minimal under G. Then X is a compact distal (point-distal) minimal set if and only if there is a point $x \in X$ such that for any homomorphism $h:X \to BC(G),h(x)$ is a right distal (weakly distal) function.References
- Jozeph Auslander and Frank Hahn, Point transitive flows, algebras of functions and the Bebutov system, Fund. Math. 60 (1967), 117–137. MR 221489, DOI 10.4064/fm-60-2-117-137
- L. Auslander and F. Hahn, Real functions coming from flows on compact spaces and concepts of almost periodicity, Trans. Amer. Math. Soc. 106 (1963), 415–426. MR 144325, DOI 10.1090/S0002-9947-1963-0144325-8
- John D. Baum, An equicontinuity condition for transformation groups, Proc. Amer. Math. Soc. 4 (1953), 656–662. MR 56286, DOI 10.1090/S0002-9939-1953-0056286-X N. Bourbaki, General topology. Parts 1, 2, Hermann, Paris; Addison-Wesley, Reading, Mass., 1966. MR 34 #5044a; MR 34 #5044b.
- Robert Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–405. MR 101283
- Robert Ellis, Equicontinuity and almost periodic functions, Proc. Amer. Math. Soc. 10 (1959), 637–643. MR 107225, DOI 10.1090/S0002-9939-1959-0107225-X
- Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
- 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
- A. W. Knapp, Distal functions on groups, Trans. Amer. Math. Soc. 128 (1967), 1–40. MR 222873, DOI 10.1090/S0002-9947-1967-0222873-3
- William A. Veech, Point-distal flows, Amer. J. Math. 92 (1970), 205–242. MR 267560, DOI 10.2307/2373504
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 32 (1972), 304-308
- MSC: Primary 54.80
- DOI: https://doi.org/10.1090/S0002-9939-1972-0288744-9
- MathSciNet review: 0288744