Ergodic theory can be described as the statistical and qualitative behavior of measurable group and semigroup actions on measure spaces. The
group is most commonly N, R,
R-+, and Z.
Ergodic theory had its origins in the work of Boltzmann in statistical mechanics problems where time- and space-distribution averages are equal. Steinhaus (1999,
pp. 237-239) gives a practical application to ergodic theory to keeping one's
feet dry ("in most cases," "stormy weather excepted") when walking
along a shoreline without having to constantly turn one's head to anticipate incoming
waves. The mathematical origins of ergodic theory are due to von Neumann, Birkhoff,
and Koopman in the 1930s. It has since grown to be a huge subject and has applications
not only to statistical mechanics, but also to number
theory, differential geometry, functional
analysis, etc. There are also many internal problems (e.g., ergodic theory being
applied to ergodic theory) which are interesting.