Minimal flows and their extensions.

*(English)*Zbl 0654.54027
North-Holland Mathematics Studies, 153; Notas de MatemĂˇtica, 122. Amsterdam etc.: North-Holland. xi, 265 p. $ 86.75; Dfl. 165.00 (1988).

This book on topological dynamics treats many of the same topics as the monograph by R. Ellis [“Lectures in topological dynamics” (1969; Zbl 0193.515)]. It keeps, however, a more dynamical viewpoint; the algebric aspects are dealt with in a separate chapter.

Certainly, the theory of equicontinuous and distal minimal flows (culminatig in Furstenberg’s structure theorem) are central in the field, and so they are presented together with proximality in great detail. One of the main emphasis of the book is put on the universal approach towards minimal flows. So the reader will find the theorems on the existence of universal minimal flows, the existence of “maximal” equicontinuous factors, disjointness, weakly mixing flows among many others. The existence of invariant measures is treated via strongly proximal flows [compare S. Glasner, “Proximal flows”, Lect. Notes Math. 517 (1976; Zbl 0322.54017)]. The book also contains Kakutani-Bebutov theorems on the representation as a flow on function spaces and a final chapter is devoted to general structure theorems, includig PI flows, for example. This well written book fills in a gap presenting recent developments in the theory.

Certainly, the theory of equicontinuous and distal minimal flows (culminatig in Furstenberg’s structure theorem) are central in the field, and so they are presented together with proximality in great detail. One of the main emphasis of the book is put on the universal approach towards minimal flows. So the reader will find the theorems on the existence of universal minimal flows, the existence of “maximal” equicontinuous factors, disjointness, weakly mixing flows among many others. The existence of invariant measures is treated via strongly proximal flows [compare S. Glasner, “Proximal flows”, Lect. Notes Math. 517 (1976; Zbl 0322.54017)]. The book also contains Kakutani-Bebutov theorems on the representation as a flow on function spaces and a final chapter is devoted to general structure theorems, includig PI flows, for example. This well written book fills in a gap presenting recent developments in the theory.

Reviewer: M.Denker

##### MSC:

54H20 | Topological dynamics (MSC2010) |

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

28D15 | General groups of measure-preserving transformations |