5. Ergodic Theory

The evolution of volumes provides useful information about dynamical systems. Conservative systems that conserve volume cannot have attractors, only dissipative systems that do not conserve volume can. Volume is but a special kind of measure and studying the behavior of more general measures provides profound insights into dynamical systems, allowing to address questions such as whether points return to the set from which they originated, whether they visit other sets, what is the averaged behavior of observables along trajectories, and what may be regarded as typical behavior. Measures are studied by measure theory which is a well-developed mathematical theory. The basis of measure theory is a measure space: a triple that consists of a basic set, a \(\sigma \)-algebra, which is the family of its measurable subsets, and a measure that assigns to each measurable subset a non-negative real number, the measure of that subset. We thus assume the state space of the dynamical system to be a measure space. To make the dynamical system compatible with the measure space one assumes that the dynamical map is a measurable function, similarly to assuming that the dynamical map is a continuous function when studying the topological properties of systems. In this chapter we thus consider the quadruple consisting of a state space, a sigma-algebra, a measure, and a family of measurable dynamical maps. Such systems are studied by ergodic theory.