Ergodic Theory

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.

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.

Petersen, K. Ergodic Theory. Cambridge, England: Cambridge University Press, 1983.

Radin, C. "Ergodic Theory." Ch. 1 in Miles of Tiles. Providence, RI: Amer. Math. Soc., pp. 17-54, 1999.

Sinai, Ya. G. Topics in Ergodic Theory. Princeton, NJ: Princeton University Press, 1993.

Smorodinsky, M. Ergodic Theory, Entropy. Berlin: Springer-Verlag, 1971.

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 237-239, 1999.

Walters, P. Ergodic Theory: Introductory Lectures. New York: Springer-Verlag, 1975.

Walters, P. Introduction to Ergodic Theory. New York: Springer-Verlag, 2000.