Ergodic Theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics.

A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set. More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important theorems are those of Birkhoff (1931) and von Neumann which assert the existence of a time average along each trajectory. The problem of metric classification of systems is another important part of the abstract ergodic theory. An outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of entropy for dynamical systems.

The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory. The underlying idea is that for certain systems the time average of their properties is equal to the average over the entire space. Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind. In geometry, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context for applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the theory of diophantine approximations, L-functions).

See also

Dynamical systems, Statistical Physics, Stochastic processes, Ergodicity, Ergodic hypothesis

Material

• Cosma’s note
• https://www.encyclopediaofmath.org/index.php/Ergodic_theory

Papers

• Eckmann, J. P., & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Reviews of modern physics, 57(3), 617.
• Bowen, R., & Ruelle, D. (1975). The ergodic theory of Axiom A flows. In The Theory of Chaotic Attractors. Springer New York, 55-76
• Pesin, Y. B. (1977). Characteristic Lyapunov exponents and smooth ergodic theory. Russian Mathematical Surveys, 32(4), 55-114.

Books

• Walters, P. (2000). An introduction to ergodic theory (Vol. 79). Springer Science & Business Media.
• Cornfeld, I. P., Fomin, S. V., & Sinai, Y. G. E. (2012). Ergodic theory (Vol. 245). Springer Science & Business Media.
• Mané, R. (2012). Ergodic theory and differentiable dynamics (Vol. 8). Springer Science & Business Media.
• Petersen, K. E. (1989). Ergodic theory (Vol. 2). Cambridge University Press.