1 - Introduction and preliminaries

Summary

Without going into the details (to which the rest of the book is devoted), we mention some of the basic questions, examples, and constructions of ergodic theory, in order to provide an indication of the content and flavor of the subject as well as to establish reference points for terminology and notation. The final section presents a few facts from measure theory and functional analysis that will be used repeatedly.

The basic questions of ergodic theory

Ergodic theory is the mathematical study of the long-term average behavior of systems. The collection of all states of a system forms a space X. The evolution of the system is represented by a transformation T: X → X, where Tx is taken as the state at time 1 of a system which at time 0 is in state x. If one prefers a continuous variable for the time, he can consider a one-parameter family {T_t: t ∈ ℝ} of maps of X into itself. When the laws governing the behavior of the system do not change with time, it is natural to suppose that T_{s + t} = T_sT_t so that {T_t: t ∈ ℝ} is a flow, or group action of U on X. A single (invertible) transformation T: X → X also determines the action of a group, namely the integers ℤ, on X.