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 $(X,\mathrm{\Sigma },\mu )$, Birkhoff’s ergodic theorem states that for any function $f\in L^1(X,\mathrm{\Sigma },\mu )$, the limit $$\bar{f}(x)\stackrel{\text{df}}{=}\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{k=0}^{n-1}f(T^k(x))$$ (the time average or the average along a trajectory) exists almost everywhere (for almost all $x\in X$). Moreover, $\bar{f}\in L^1(X,\mathrm{\Sigma },\mu )$, and if $\mu (X)<\mathrm{\infty }$, then $${\int }_{X}f\mathrm{d}\mu ={\int }_X\bar{f}\mathrm{d}\mu .$$

For a measurable flow $(T_t{)}_{t\ge 0}$ in a $\sigma$-finite measure space $(X,\mathrm{\Sigma },\mu )$, Birkhoff’s ergodic theorem states that for any function $f\in L^1(X,\mathrm{\Sigma },\mu )$, the limit $$\bar{f}(x)\stackrel{\text{df}}{=}\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}f(T_t(x))\mathrm{d}t$$ 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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