The ergodic approach

Abstract

In previous chapters we discussed dynamical systems mainly from a geometrical or topological point of view. The geometrical approach is intuitively appealing and lends itself to suggestive graphical representations. Therefore, it has been tremendously successful in the study of low-dimensional systems: continuous-time systems with one, two or three variables; discrete-time systems with one or two variables. For higher-dimensional systems, however, the approach has encountered rather formidable obstacles and rigorous results and classifications are few. Thus, it is sometimes convenient to change perspective and adopt a different approach, based on the concept of measure, and aimed at the investigation of the statistical properties of ensembles of orbits. This requires the use and understanding of some basic notions and results, to which we devote this chapter. The ergodic theory of dynamical systems often parallels its geometric counterpart and many concepts discussed in chapters 3–8, such as invariant, indecomposable and attracting sets, attractors, and Lyapunov characteristic exponents will be reconsidered in a different light, thereby enhancing our understanding of them. We shall see that the ergodic approach is very powerful and effective for dealing with basic issues such as chaotic behaviour and predictability, and investigating the relationship between deterministic and stochastic systems. From the point of view of ergodic theory, there is no essential difference between discrete- and continuous-time dynamical systems. Therefore, in what follows, we develop the discussion mostly in terms of maps, mentioning from time to time special problems occurring for flows.