Abstract
In this final chapter we take up some loose ends of the previous chapters. First of all, we prove that every extension of compact minimal flows can be ‘lifted’ (by means of proximal extensions—even strongly proximal ones) to a RIM-extension of compact minimal flows. As proximal extensions are, from a dynamical point of view, close to isomorphisms (asymptotically, different points in any fiber cannot be distinguished), one expects that useful properties of these RIM-liftings throw their shadows on the original extensions—provided additional conditions of the latter can be lifted to the former. This procedure has been used in Chapter V in the first half of the proof of FST: a distal extension of compact minimal flows lifts to an open RIM-extension; if the former has no non-trivial equicontinuous factors then the same is true for the latter; hence the latter is weakly mixing and, consequently, the former is weakly mixing. The other half of the proof is that ‘weakly mixing’ can be improved to ‘isomorphisms’ for distal extensions. In order to generalize this procedure we study such ‘shadow-diagrams’ in more detail in Section 1. As one more application, we show that for distal extensions of compact minimal flows the relative regionally proximal relation is an equivalence relation (i.e., Q∅ = S e∅ for such ∅). Exactly the same arguments as for distal extensions can be used for a larger class of extensions, the socalled relatively incontractible (RIC) extensions, which can be characterized as the extensions which are ‘orthogonal to’ or ‘disjoint from’ all proximal extensions (cf. Section 2).
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Notes
(7.1) Notes to Section 1
Here we follow S. Glasner [1975a].
Introduced in S. Glasner [1975a] as a relativization of the notion of a strongly proximal flow (see also Note 3 below).
Introduced in S. Glasner [1975b]. The motivation of this notion lies in the study of so-called “boundaries” of connected semi-simple Lie groups with finite center (cf. H. Furstenberg [1963a]). If G= KAN is the Iwasawa decomposition of such a group and P is the normalizer of N in G then, under the natural action of G, G/ P is a compact minimal flow which is (in our terminology) strongly proximal. This result, due to C.C. Moore [1964a,b], is reproduced, among others, in G. Warner [1972], Prop. 1.2.3.13, and in J. de Vries [1982].
In [Wo], p. 223 a diagram is constructed such that the conditions (a) and (b) are satisfied for what are called there ‘RMM-extensions’ (RIC-extensions and open RIMextensions are RMM, as are B-extensions which have a RIM).
In this form this result was proved in [Wo], VII.3.22a, using a generalization of the methods developed in D.C. McMahon [1978]. The proof that I present was communicated to me by Jaap van der Woude.
(7.2) Notes to Section 2
The notion of an incontractible compact minimal flow was introduced in S. Glasner [1975b]. He used the characterization given in (2.9)1 as a definition.
See e.g. [Gl], X.1.3.
This theorem follows easily from the corresponding result for B-extensions: cf. (3.19) and also Note 10 in (7.3) ahead.
Like theorem (2.12), also this theorem follows from the corresponding result for B-extensions: cf. (3.20), and also Note 11 in (7.3) ahead.
See R. Ellis, S. Glasner & L. Shapiro [19751.
(7.3) Notes to Section 3
This type of diagram was described in P.S. Schoenfeld [1974]; the metric case is due to W.A. Veech [1970]. See also J. Auslander & S. Glasner [1977].
Introduced by J. Auslander, see P.S. Schoenfeld [1974] and J. Auslander & S. Glasner [1977]. Most of our results on highly proximal extensions (including those in (5.15)) can be found in one of these publications.
In view of IV(2.18) this requires T to be non-Abelian. For an example with Abelian T, see D.C. Mcmahon [1976a] (there a proximally equicontinuous compact minimal flow is constructed which is not locally almost periodic; in view of V(6.1) 2,4 this provides a proximal extension—of its largest equicontinuous factor—which is not highly proximal).
Yet another proof: by [Wi], 14.2.3 every open irreducible map onto a Hausdorff space is a homeomorphism.
In this form the A G-diagram appears essentially in W.A. Veech [1970].
See [Wo], IV.3.12.
In S. Glasner [1975b] (Lemma 5.2) a special case is proved, namely, that an almost 1:1 extension of compact minimal flows lifts strong proximality of flows. For our Proposition, see [Wo], VII.1.9(b).
Cf. [Wo], IV.4.5 where a more general situation is considered.
For the case that X and Y are metrizable this result was proven in D.C. Mcmahon [1976]. In its fullest generality this result appears in W.A. Veech [19771b].
Unfortunately, this approach was discovered too late to incoporate it more elegantly in the text.
(7.4) Notes to Section 4
See [Wo], IV 3.13. It follows that ‘open + point-distal B’ for extensions of compact minimal flows; this was proved in R. Ellis [1973].
In the absolute case there is a similar result in D.C. McMahon & T.-S. Wu [1972], Thm. 1 (however, there T is assumed to be Abelian).
This is Prop. 1.3 in R. Ellis [1973].
Introduced and studied in R. Ellis, S. Glasner & L. Shapiro [1975]. For an ‘intrinsic’ characterization, see (4.37); cf. also Note 17 below.
Cf. P.S. Shoenfeld [1974] and J. Auslander & S. Glasner [1977]. See also Section IV.5 in [Wo]. (Our terminology slightly deviates from that in W.A. VEECH [1970]: what we call a strictly-Al-extension is called there an ‘AI-extension’). For an intrinsic characterization, see J.C.S.P. van der Woude [1985].
Some of the tedious details in the proof can be avoided by working from the outset with ambits and base point preserving extensions. But we prefer the base pointfree approach.
In this form the result appears in W.A. Veech [1977b], though the construction of canonical PI-towers appears already in R. Ellis, S. Glasner & L. Shapiro [1975].
See [Wi], 14.2.1 (here an irreducible mapping is called ‘minimal’).
The absolute case is, essentially, in Proposition 8.10 of R. Ellis, S. Glasner & L. Shapiro [1975].
For the existence and unicity of the universal strongly proximal extension of a compact minimal flow, see S. Glasner [1975a].
For a proof based on the characterization in (3.3)(ii), see [Wo], p. 109.
In W.A. Veech [1970], p. 241 this question is left open, even in the absolute case.
Left open in [Wo].
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de Vries, J. (1993). Structure of Extensions. In: Elements of Topological Dynamics. Mathematics and Its Applications, vol 257. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8171-4_6
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