Birkhoff ergodic theorem

This category corresponds roughly to MSC {{{id}}} {{{title}}}; see {{{id}}} at MathSciNet and {{{id}}} at zbMATH.

One of the most important theorems in ergodic theory. For an endomorphism $ T $ of a $ \sigma $-finite measure space $ (X,\Sigma,\mu) $, Birkhoff’s ergodic theorem states that for any function $ f \in {L^{1}}(X,\Sigma,\mu) $, the limit $$ \overline{f}(x) \stackrel{\text{df}}{=} \lim_{n \to \infty} \frac{1}{n} \sum_{k = 0}^{n - 1} f \! \left( {T^{k}}(x) \right) $$ (the time average or the average along a trajectory) exists almost everywhere (for almost all $ x \in X $). Moreover, $ \overline{f} \in {L^{1}}(X,\Sigma,\mu) $, and if $ \mu(X) < \infty $, then $$ \int_{X} f ~ \mathrm{d}{\mu} = \int_{X} \overline{f} ~ \mathrm{d}{\mu}. $$

For a measurable flow $ (T_{t})_{t \geq 0} $ in a $ \sigma $-finite measure space $ (X,\Sigma,\mu) $, Birkhoff’s ergodic theorem states that for any function $ f \in {L^{1}}(X,\Sigma,\mu) $, the limit $$ \overline{f}(x) \stackrel{\text{df}}{=} \lim_{t \to \infty} \frac{1}{t} \int_{0}^{t} f({T_{t}}(x)) ~ \mathrm{d}{t} $$ exists almost everywhere, with the same properties as $ f $.

Birkhoff’s theorem was stated and proved by G.D. Birkhoff  . It was then modified and generalized in various ways (there are theorems that contain, in addition to Birkhoff’s theorem, also a number of statements of a somewhat different kind, which are known in probability theory as ergodic theorems (cf. Ergodic theorem); there also exist ergodic theorems for more general semi-groups of transformations  ). Birkhoff’s ergodic theorem and its generalizations are known as individual ergodic theorems, since they deal with the existence of averages along almost each individual trajectory, as distinct from statistical ergodic theorems — the von Neumann ergodic theorem and its generalizations. (In non-Soviet literature, the term “pointwise ergodic theorem” is often used to stress the fact that the averages are almost-everywhere convergent.)

References

	G.D. Birkhoff, “Proof of the ergodic theorem”, Proc. Nat. Acad. Sci. USA, 17 (1931), pp. 656–660. MR Template:ZBL Template:ZBL
	A.B. Katok, Ya.G. Sinai, A.M. Stepin, “Theory of dynamical systems and general transformation groups with invariant measure”, J. Soviet Math., 7: 6 (1977), pp. 974–1065; Itogi Nauk. i Tekhn. Mat. Analiz, 13 (1975), pp. 129–262. MR0584389 Template:ZBL

Comments

In non-Soviet literature, the term “mean ergodic theorem” is used instead of “statistical ergodic theorem”.

A comprehensive overview of ergodic theorems is found in  . Many books on ergodic theory contain full proofs of (one or more) ergodic theorems; see e.g.  .

References

	U. Krengel, “Ergodic theorems”, de Gruyter (1985). MR0797411 Template:ZBL
	K. Peterson, “Ergodic theory”, Cambridge Univ. Press (1983). MR0833286 Template:ZBL