Birkhoff ergodic theorem

2020 Mathematics Subject Classification: Primary: 37A30 Secondary: 37A0537A10 [MSN][ZBL]

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 [B]. 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 [KSS]). 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.)

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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 [K]. Many books on ergodic theory contain full proofs of (one or more) ergodic theorems; see e.g. [P].

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