Proximality in minimal transformation groups

Shapiro, Leonard

doi:10.1090/S0002-9939-1970-0266183-2

Proximality in minimal transformation groups
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by Leonard Shapiro

Proc. Amer. Math. Soc. 26 (1970), 521-525

DOI: https://doi.org/10.1090/S0002-9939-1970-0266183-2

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Abstract:

We present an action of the integers on a compact metric space such that in the resulting minimal topological transformation group the proximal relation is an equivalence relation but is not closed.

References

Joseph Auslander, On the proximal relation in topological dynamics, Proc. Amer. Math. Soc. 11 (1960), 890–895. MR 164335, DOI 10.1090/S0002-9939-1960-0164335-7
Joseph Auslander, Endomorphisms of minimal sets, Duke Math. J. 30 (1963), 605–614. MR 155311
Jesse Paul Clay, Proximity relations in transformation groups, Trans. Amer. Math. Soc. 108 (1963), 88–96. MR 154269, DOI 10.1090/S0002-9947-1963-0154269-3
Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494
Harvey B. Keynes, On the proximal relation being closed, Proc. Amer. Math. Soc. 18 (1967), 518–522. MR 212780, DOI 10.1090/S0002-9939-1967-0212780-X

The structure of $H$-cascades

William A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem $\textrm {mod}\ 2$, Trans. Amer. Math. Soc. 140 (1969), 1–33. MR 240056, DOI 10.1090/S0002-9947-1969-0240056-X