×

Topological dynamix. (Repr. of diss.). (English) Zbl 0654.54026

CWI Tracts, 22. Centrum voor Wiskunde en Informatica. Amsterdam: Mathematisch Centrum. X, 298 p. Dfl. 44.90 (1986).
This monograph is the author’s doctoral dissertation written at the Free University of Amsterdam. It consists of a detailed study of a number of topics in the abstract theory of topological dynamics related to the structure of minimal flows and presents the “state of the art” at the time of writing (1982). It is for the most part self-contained; thus earlier results are woven together with the author’s own contributions.
Chapters I and II are introductory, establish notation and terminology, and indicate the author’s point of view. A topological transformation group, (“ttg”) or “flow” is a continuous (left) action of a topological group T on a topological space X. It is usually assumed that T is discrete and X is compact Hausdorff. If (T,X) and (T,Y) are ttg’s a homomorphism is a continuous equivariant map \(\phi\) : \(X\Rightarrow Y\); in this case Y is a “factor” of X and X is an “extension” of Y. A ttg is point transitive if there is a point with dense orbit and minimal if every orbit is dense. There is a naturally defined action of T on its Čech compactification \(\beta\) T; (T,\(\beta\) T) is universal in that every point transitive ttg is a factor and every minimal subset M of \(\beta\) T is a universal minimal ttg.
The determination of the equicontinuous structure relation of a minimal ttg is a major theme. Its study is initiated in Chapter III, and completed in Chapters VII and VIII. It is elementary that every ttg has a maximal equicontinuous factor and that the equivalence relation determining this factor contains the regionally proximal relation \(Q=\cap [\overline{T\alpha}| \alpha \in {\mathcal U}_ X]\) indeed it is the smallest closed invariant equivlence relation containing Q. This can be relativized: if \(\phi\) : \(X\Rightarrow Y\) is a homomorphism and \(Q_{\phi}=\cap [\overline{T\alpha \cap R_{\phi}}| \alpha \in {\mathcal U}_ X],\) and \(E_{\phi}\) the smallest closed invariant equivalence relation containing \(Q_{\phi}\), then \(X/E_{\phi}\) is the maximal almost periodic (equi-continuous) extension of Y which can be interpolated in \(\phi\) : \(X\Rightarrow Y\). Remarkably, for minimal flows the relations \(Q_{\phi}\) and \(E_{\phi}\) frequently coincide. This is the case for RIC (relatively incontractible) extensions (a strong form of openness) the Bronstein condition (almost periodic points dense in the fibered product) and open RIM (relatively invariant measure) extensions (the existence of a measure structure on the fibers, which is the absolute case reduces to the existence of an invariant measure on X). The proofs of these and related facts are difficult and involved and require the development of some intricate machinery, in particular the \({\mathcal F}\) topologies. These are “weak” topologies on certain subsets of the space (introduced by Furstenberg for his study of distal minimal flows and characterized in a different way by Ellis and his collaborators).
Another central theme is that of quasifactor. If a group T acts on a space X, there is a naturally defined action on \(2^ X\) the space of closed subsets of X; a quasifactor of (T,X) is a minimal subset of \((T,2^ X)\). Quasifactors are useful for the study of certain dynamical notions (for example the \({\mathcal F}\) topologies can be defined using them) and are also of intrinsic interest. The maximally highly proximal (MHP) flows are quasifactors of the universal minimal ttg M; they are defined in terms of certain subsets of M (MHP generators) which have strong algebraic properties. The MHP flows are characterized as those flows which are open images of M.
Disjointness relations are considered in depth. Disjointness is a kind of independence condition which allows for a global consideration of flows and homomorphisms. Two minimal ttgs (T,X) and (T,Y) are disjoint if the product ttg (T,X\(\times Y)\) is minimal, and two homomorphisms of minimal ttg’s (with the same codomain) are disjoint if the fibered product is minimal. If \({\mathcal D}\) and \({\mathcal W}{\mathcal M}\) denote the class of distal and weakly mixing flows respectively, and \(\perp\) indicates disjointness, then we have (if the group T is strongly amenable) \({\mathcal D}={\mathcal W}{\mathcal M}={\mathcal W}{\mathcal M}^{\perp \perp}\) and \({\mathcal D}^{\perp \perp}={\mathcal W}{\mathcal M}^{\perp}\). A useful table of such disjointness relations is included. Disjointness is also related to relative primeness (no common factor) and a number of conditions are given where the two notions are equivalent. Another interesting result is that a homomorphism of minimal ttgs is open if and only if it is disjoint from every highly proximal extension of the codomain. A variant is weak disjointness - “minimal” is replaced by “point transitive” in the definitions and is related to disjointness of the maximal equicontinuous factors.
This work builds on contributions of Ellis, Furstenberg, Veech, (in particular the important survey article by W. A. Veech in Bull. Am. Math. Soc. 83, 775-830 (1977; Zbl 0384.28018)) Glasner, and McMahon. Parts of Chapters V, VII and VIII are essentially the contents of the author’s joint papers with the reviewer [Ergodic Theory Dyn. Syst. 1, 389-412 (1981; Zbl 0489.54039)] and both with D. C. McMahon and T. S. Wu [Ergodic theory Dyn. Syst. 4, 323-351 (1984; Zbl 0559.22004)]. The style is somewhat encyclopedic and it isn’t always easy to locate results; perhaps it would have been possible to present a unified treatment of some topics which are spread over several chapters. The author’s prose is colorful if occasionally awkward; his predilection for puns is exemplified by the title.
Subsequent to the completion of his dissertation, the author continued to work in topological dynamics (partly in collaboration with Wu and McMahon). Unfortunately, he was not to obtain a position at a mathematics department in the Netherlands, and is no longer working in this area. Nevertheless, the work under review represents a substantial achievement.
Reviewer: J.Auslander

MSC:

54H20 Topological dynamics (MSC2010)
54-02 Research exposition (monographs, survey articles) pertaining to general topology