Ergodic sets

Introduction. Ergodic sets were introduced by Kryloff and Bogoliouboff in 1937 in connection with their study of compact dynamical systems The purpose of this paper is to review some of the work that has since been done on the theory that centers around this notion, and to present a number of supplementary remarks, applications, and simplifications. For simplicity we shall confine attention to systems with a discrete time. Continuous flows present no difficulty, but the development of a corresponding theory for general transformation groups is still in an incomplete stage. An example due to Kolmogoroff (see