Chapter 10 On the interplay between measurable and topological dynamics

Abstract

This chapter discusses the interplay between measurable and topological dynamics. Ergodic theory or measurable dynamics and topological dynamic are the two sister branches of the theory of dynamical systems. The simplest dynamical systems are the periodic ones. In the absence of periodicity, the crudest approximation to this is approximate periodicity. In the topological setup, there is no such convenient decomposition describing the system in terms of its indecomposable parts and one has to use some less satisfactory substitutes. Natural candidates for indecomposable components of a topological dynamical system are the “orbit closures” (i.e. the topologically transitive subsystems) or the “prolongation” cells that often coincide with the orbit closures. The minimal subsystems are of particular importance in this chapter. Although the study of the general system can be reduced to that of its minimal components, the analysis of the minimal systems is nevertheless an important step toward a better understanding of the general system.