𝐹-minimal sets

Markley, Nelson G.

doi:10.1090/S0002-9947-1972-0284997-6

$F$-minimal sets
HTML articles powered by AMS MathViewer

by Nelson G. Markley PDF

Trans. Amer. Math. Soc. 163 (1972), 85-100 Request permission

Abstract:

An $F$-minimal set is the simplest proximal extension of an equicontinuous minimal set. It has one interesting proximal cell and all the points in this proximal cell are uniformly asymptotic. The Sturmian minimal sets are the best known examples of $F$-minimal sets. Our analysis of them is in terms of their maximal equicontinuous factors. Algebraically speaking $F$-minimal sets are obtained by taking an invariant *-closed algebra of almost periodic functions and adjoining some suitable functions to it. Our point of view is to obtain these functions from the maximal equicontinuous factor. In §3 we consider a subclass of $F$-minimal sets which generalize the classical Sturmian minimal sets, and in §4 we examine the class of minimal sets obtained by taking the minimal right ideal of an $F$-minimal set and factoring by a closed invariant equivalence relation which is smaller than the proximal relation.

References

Joseph Auslander, Endomorphisms of minimal sets, Duke Math. J. 30 (1963), 605–614. MR 155311
Ethan M. Coven and Michael S. Keane, The structure of substitution minimal sets, Trans. Amer. Math. Soc. 162 (1971), 89–102. MR 284995, DOI 10.1090/S0002-9947-1971-0284995-1
Robert Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc. 94 (1960), 272–281. MR 123636, DOI 10.1090/S0002-9947-1960-0123636-3
Robert Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc. 94 (1960), 258–271. MR 123635, DOI 10.1090/S0002-9947-1960-0123635-1
E. E. Floyd, A nonhomogeneous minimal set, Bull. Amer. Math. Soc. 55 (1949), 957–960. MR 33535, DOI 10.1090/S0002-9904-1949-09318-7
Richard Freiman, Regular minimal sets over the circle and the Ellis minimal set, Math. Systems Theory 6 (1972), 145–163. MR 307214, DOI 10.1007/BF01706086
W. H. Gottschalk, Characterizations of almost periodic transformation groups, Proc. Amer. Math. Soc. 7 (1956), 709–712. MR 79754, DOI 10.1090/S0002-9939-1956-0079754-6
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
Gustav A. Hedlund, Sturmian minimal sets, Amer. J. Math. 66 (1944), 605–620. MR 10792, DOI 10.2307/2371769
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144

Continuous groups

17

34

W. A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), 719–751. MR 187014, DOI 10.2307/2373071
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
Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491