Lectures on topological dynamics.

*(English)*Zbl 0193.51502
New York: W. A. Benjamin, Inc., xv, 211 p. (1969).

Topological dynamics is an abstract theory of dynamical systems. It is concerned with the study of transformation groups with respect to those topological properties whose prototype occurred in the classical dynamics. This book is devoted to a self-contained presentation of the more recent results in topological dynamics, most of which are due to the author.

“The central theme is the classification of minimal sets. Although there remains much to be done, I felt it would be useful to present a unified account of the results obtained thus far. Recent work on the abstract theory indicates first, that the proper approach is to view a given minimal set as a “point” in the collection of all minimal sets rather than as an isolated object, and second that the abstract theory is closer in spirit to functional analysis than it is to differential equations. These two points are emphasized in the text” (from the author’s Introduction).

In the first six chapters the basic notions (transformation group, minimal set, enveloping semigroup, distal, proximal) are introduced. In Chapters 7 and 8 the universal point transitive (p.t.) transformation group \((\beta T,T)\) associated with a discrete group \(T\) is considered. Since every point transitive transformation group \((X,T)\) is a homomorphic image of \((\beta T,T)\), \((X,T)=(\beta T/R,T)\), where \(R\) is an invariant closed equivalence relation on \(\beta T\). By a classical result of Stone, the study of p.t. transformation groups is equivalent to the study of certain subalgebras of the Banach algebra \(\mathcal C(\beta T)\). A representation of p.t. transformation groups is given in Chapter 9. The next chapter is devoted to a description of subalgebras which correspond to minimal sets. In Chapter 11 a group is associated with every minimal set. These groups are used in chapters 12, 13, 14 to study distal and almost periodic extensions of minimal sets. A Galois theory of distal extensions is presented in Chapter 15 and applied to obtain a generalisation of Furstenberg’s structure theorem for distal minimal flows. Chapter 16 is devoted to an exposition on Knapp’s work on Fourier analysis of almost periodic group extensions of a minimal set \(X\). A vector bundle over \(X\) is associated with every unitary representation of the structure group. The set of all such vector bundles is described in Chapter 17. Chapter 18 contains some results on disjointness of minimal sets. At the end of each chapter there is a section which includes references, comments and unsolved problems.

The book is self-contained, complete proofs are given.

“The central theme is the classification of minimal sets. Although there remains much to be done, I felt it would be useful to present a unified account of the results obtained thus far. Recent work on the abstract theory indicates first, that the proper approach is to view a given minimal set as a “point” in the collection of all minimal sets rather than as an isolated object, and second that the abstract theory is closer in spirit to functional analysis than it is to differential equations. These two points are emphasized in the text” (from the author’s Introduction).

In the first six chapters the basic notions (transformation group, minimal set, enveloping semigroup, distal, proximal) are introduced. In Chapters 7 and 8 the universal point transitive (p.t.) transformation group \((\beta T,T)\) associated with a discrete group \(T\) is considered. Since every point transitive transformation group \((X,T)\) is a homomorphic image of \((\beta T,T)\), \((X,T)=(\beta T/R,T)\), where \(R\) is an invariant closed equivalence relation on \(\beta T\). By a classical result of Stone, the study of p.t. transformation groups is equivalent to the study of certain subalgebras of the Banach algebra \(\mathcal C(\beta T)\). A representation of p.t. transformation groups is given in Chapter 9. The next chapter is devoted to a description of subalgebras which correspond to minimal sets. In Chapter 11 a group is associated with every minimal set. These groups are used in chapters 12, 13, 14 to study distal and almost periodic extensions of minimal sets. A Galois theory of distal extensions is presented in Chapter 15 and applied to obtain a generalisation of Furstenberg’s structure theorem for distal minimal flows. Chapter 16 is devoted to an exposition on Knapp’s work on Fourier analysis of almost periodic group extensions of a minimal set \(X\). A vector bundle over \(X\) is associated with every unitary representation of the structure group. The set of all such vector bundles is described in Chapter 17. Chapter 18 contains some results on disjointness of minimal sets. At the end of each chapter there is a section which includes references, comments and unsolved problems.

The book is self-contained, complete proofs are given.

Reviewer: I. U. Bronshtein