Birkhoff's Ergodic Theorem

Let T be an ergodic endomorphism of the probability space X and let f:X->R be a real-valued measurable function. Then for almost every x in X, we have

 1/nsum_(j=1)^nf degreesT^j(x)->intfdm

as n->infty. To illustrate this, take f to be the characteristic function of some subset A of X so that

 f(x)={1   if x in A; 0   if x not in A.

The left-hand side of (1) just says how often the orbit of x (that is, the points x, Tx, T^2x, ...) lies in A, and the right-hand side is just the measure of A. Thus, for an ergodic endomorphism, "space-averages = time-averages almost everywhere." Moreover, if T is continuous and uniquely ergodic with Borel measure m and f is continuous, then we can replace the almost everywhere convergence in (1) with "everywhere."

See also

Birkhoff's Theorem, Ergodic Theory

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Cornfeld, I.; Fomin, S.; and Sinai, Ya. G. Appendix 3 in Ergodic Theory. New York: Springer-Verlag, 1982.

Referenced on Wolfram|Alpha

Birkhoff's Ergodic Theorem

Cite this as:

Weisstein, Eric W. "Birkhoff's Ergodic Theorem." From MathWorld--A Wolfram Web Resource.

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