Abstract
As in the previous chapter, also in this chapter T will always denote an arbitrary topological group. The notation of Chapter IV will be used throughout. In particular, S T denotes the universal enveloping semigroup—so (L T , e) is the universal ambit—and J0 is the set of all idempotents of ST that are situated in a minimal left ideal.
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Notes
(7.1) Notes to Section 1
Some authors define a compact flow X to be weakly mixing whenever 2f=IXI; see e.g. [Br], 3.13.14. Others call a compact flow weakly mixing whenever S%- = XXX (i.e., whenever the flow has no non-trivial equicontinuous factor); see e.g. R. Ellis [1981]. In W.A. Veech [1977b] our definition is used, but there the notion is called ‘topologically transitive’. See also Notes 4 and 5 in IV(7.1). By Theorem (1.19), the two alternative definitions mentioned above agree with our definition for compact minimal flows that admit an invariant probability measure (for all compact minimal flows when T is amenable). By VI(3.19) and VI(3.20) ahead, the definitions also agree for compact minimal flows usuell that Xhas a dense set of almost periodic points.
In many publications, S % is denoted by E% (or just E); see e.g. [Wo].
Cf. R. Ellis & W.H. Gottschalk [I960].
Actually, the distal case is a special case of that of an invariant measure: every distal compact minimal flow has an invariant measure; cf. (6.5) 6.
See [Wo], VII.3.22.
See [Wo], VII.3.11. For the proper context of this result, see Note 14 below.
See [Wo], VII.3.17.
This example occurs in D.C. McMahon [1976].
See R. Ellis & H.B. Keynes [1971]; this result follows easily from VI(3.20). For a rather simple proof, using (2.14)3, see D.C. McMahon & T.-S. Wu [1980b].
Under the additional assumption that X is metrizable this result follows from H.B. Keynes & J.B. Robertson [1969a].
(7.2) Notes to Section 2
See [El], 5.22 and [El], 14.2 for the absolute case of 2 and for 3, respectively. (The relative case of 2 is an obvious modification of the absolute case.)
For additional conditions, guaranteeing that a composition of equicontinuous extensions is again equicontinuous, see D.C. McMahon, J. van der Woude & T.-S. Wu [1987]. Here earlier results of R.J. Sacker & G.R. Sell [1974] are generalized. An interesting counter example can be found in D.C. McMahon & T.-S. Wu [1976].
It can also be described in terms of function algebras (i.e., compact minimal flows viewed as minimal ambits); see also the final observation in VI(5.6).
For other cases, consult the references in the Notes 7 and 8 in (7.1). See also Note 11 in VI(7.3) ahead.
For other cases, consult the references in the Notes 9 and 10 in (7.1). See also Note 10 in VI(7.3) ahead.
Cf. Lemma 2.3 in D.C. McMahon & T.-S. Wu [1983]. The absolute case is in D.C. McMahon & T.-S. Wu [1980b], Prop. 3.3.
(7.3) Notes to Section 3
Due to D.C. McMahon & T.-S. Wu [1981]. The conclusions of (3.8) and (3.12) are not contained in their paper. In stead of (3.12) they proved that <j> K is open and point-distal for suitable K.
See S. Glasner [1975]. This definition doesn’t say what a RIM is, only when a extension has a RIM. See (6.5) 1.
This is a part of Corollary (1.9) in D.C. McMahon [1978] (the non-metric case). My presentation has been influenced by remarks of T.-S. Wu, communicated to me by Jaap van der Woude.
It follows that a minimally almost periodic group T (like e.g. SL(2,U)) admits no non-trivial distal minimal flows: use IV(3.38).
(7.4) Notes to Section 4
See for example W. Ruppert [1984], 1.3.15.2; this is a rather simple example but has the is disadvantage that it is not clear whether it has the form S T for some group T. For more complicated examples, see J.W. Baker & P. Milnes [1977]: if T is an infinite discrete group then in ßT (~S T : see IV(5.22)) the ‘maximal’ subgroups uM (M a minimal left ideal, ueJ(M)) are not closed. (Also, the canonical isomorphisms between the groups that partition M need not be homeomorphisms.)
See R. Ellis [1967, 1968] and also Note 5 in (7.2).
Essentially, this is Proposition 29 of R. Ellis [1967]. Our Theorem (4.19) is, essentially, Proposition 27 in this paper. In this paper, Ellis calls a ‘group extension’ what we call a ‘weak group extension’; he defines it as an extension satisfying condition (4.19) (ii).
In general, the converse is not true. The first counter example was given in A.W. Knapp [1968] (for a finite, noncommutative group T). For Abelian T7, several results in positive direction were obtained in R. Ellis, S. Glasner & L. Shapiro [1976], but in D. Rudolph [1979] a counter example occurs with non-Abelian T. In S. Glasner & B. Weiss [1983] counter examples are given with T = U and with T = Z (certain horocycle flows).
This is minor modification of [El], 18.7: there it is required that TxCuX (this clearly implies that uX is dense); see also the proofs in (4.34) 1.
(7.5) Notes to Section 5
The term ‘r-topology’ occurs in R. Ellis [1969]. Our method of introducting these topologies is taken from [Gl]; originally, this method was developed in R. Ellis, S. Glasner & L. Shapiro [1975]. More about the possible définitions of T-topologies will be said in Note 14 below.
See [Du], III.5.1.
See R. Ellis [1967]; here another (but equivalent: see R. Ellis, S. Glasner & L. Shapiro [1975]) definition of T-topology was used.
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© 1993 Springer Science+Business Media Dordrecht
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de Vries, J. (1993). Equicontinuity and Distality. In: Elements of Topological Dynamics. Mathematics and Its Applications, vol 257. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8171-4_5
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