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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. Its mathematical origins 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.
See also Ambrose-Kakutani Theorem, Birkhoff's Ergodic Theorem, Dye's Theorem, Dynamical System, Hopf's Theorem, Ornstein's Theorem
References
Billingsley, P. Ergodic Theory and Information. New York: Wiley, 1965.
Cornfeld, I.; Fomin, S.; and Sinai, Ya. G. Ergodic Theory. New York: Springer-Verlag, 1982.
Katok, A. and Hasselblatt, B. An Introduction to the Modern Theory of Dynamical Systems.
Cambridge, England: Cambridge University Press, 1996.
Nadkarni, M. G. Basic Ergodic Theory. India: Hindustan Book Agency, 1995.
Parry, W. Topics in Ergodic Theory. Cambridge, England: Cambridge University Press, 1982.
Smorodinsky, M. Ergodic Theory, Entropy. Berlin: Springer-Verlag, 1971.
Walters, P. Ergodic Theory: Introductory Lectures. New York: Springer-Verlag, 1975.