Ergodic theory lies at the intersection of many areas of mathematics, including smooth dynamics, statistical mechanics, probability, harmonic analysis, and group actions. Problems, techniques, and results are related to many other areas of mathematics, and ergodic theory has had applications both within mathematics and to numerous other branches of science. Ergodic theory has particularly strong overlap with other branches of dynamical systems; to clarify what distinguishes it from other areas of dynamics, we start with a quick overview of dynamical systems.

Dynamical systems is the study of systems that evolve with time. The evolution of a dynamical system is given by some fixed rule that determines the states of the system a short time into the future, given only the present states. Reflecting the origins of the subject in celestial mechanics, the set of states through which the system evolves with time is called an orbit. Many important concepts in dynamical systems are related to...