Topological dynamics / Walter Helbig Gottschalk, Gustav Arnold Hedlund

Bibliogr. p. 159-164. Index

In part I, which consists of 11 sections, various concepts and theorems of dynamical systems are generalized to suitably extensive classes of topological transformation groups. Isomorphisms, homomorphisms, orbits, orbit closures, invariant subsets, etc., of topological transformation groups are defined, as are partitions and decompositions of spaces. Theorems are given stating when the orbit closures constitute a partition of X. ... Part I is written in a very precise terse style, and definitions and theorems are given under most general conditions. It contains a fully organized presentation of topological dynamics from a certain point of view.
Part II consists of three sections dealing with some specific examples of flows and has a much more classical approach to the subject. Sections 12 and 13 are respectively on symbolic dynamics and geodesic flows, subjects with which one of the authors has been intimately associated. The theorem on the topological transitivity of a geodesic flow on surfaces of constant negative curvature is proved. Section 14 is on cylinder homeomorphisms.
The book is a considerable technical achievement for the authors in their efficient and neat organization of the material. The reviewer regrets however the restrictive aims of the authors which lead to the omission of the mention of some recent progress in the theory where essentially new techniques were introduced, for example, the Gelfand and Fomin approach to the study of geodesic flows (MathSciNet)