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[[Category:Ergodic theorems, spectral theory, Markov operators]]
 
[[Category:Ergodic theorems, spectral theory, Markov operators]]
  
One of the most important theorems in [[Ergodic theory|ergodic theory]]. For an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b0165301.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b0165302.png" /> with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b0165303.png" />-finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b0165304.png" /> Birkhoff's ergodic theorem states that for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b0165305.png" /> the limit
+
One of the most important theorems in [[Ergodic theory|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
 +
$$
 +
\int_{X} f ~ \mathrm{d}{\mu} = \int_{X} \overline{f} ~ \mathrm{d}{\mu}.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b0165306.png" /></td> </tr></table>
+
For a [[Measurable flow|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 $.
  
(the time average or the average along a trajectory) exists almost everywhere (for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b0165307.png" />). Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b0165308.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b0165309.png" />, then
+
Birkhoff’s theorem was stated and proved by G.D. Birkhoff {{Cite|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|Ergodic theorem]]); there also exist ergodic theorems for more general semi-groups of transformations {{Cite|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|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.)
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b01653010.png" /></td> </tr></table>
 
 
 
For a [[Measurable flow|measurable flow]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b01653011.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b01653012.png" /> with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b01653013.png" />-finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b01653014.png" /> Birkhoff's ergodic theorem states that for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b01653015.png" /> the limit
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b01653016.png" /></td> </tr></table>
 
 
 
exists almost everywhere, with the same properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b01653017.png" />.
 
 
 
Birkhoff's theorem was stated and proved by G.D. Birkhoff {{Cite|B}}. It was then modified and generalized in various ways (there are theorems which 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|Ergodic theorem]]); there also exist ergodic theorems for more general semi-groups of transformations {{Cite|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|von Neumann ergodic theorem]] and its generalizations. (In non-Soviet literature the term "pointwise ergodic theorempointwise ergodic theorem" is often used to stress the fact that the averages are almost-everywhere convergent.)
 
  
 
====References====
 
====References====
 
{|
 
{|
|valign="top"|{{Ref|B}}|| G.D. Birkhoff, "Proof of the ergodic theorem" ''Proc. Nat. Acad. Sci. USA'' , '''17''' (1931) pp. 656–660 {{MR|}} {{ZBL|0003.25602}} {{ZBL|57.1011.02}}
+
|valign="top"|{{Ref|B}}|| G.D. Birkhoff, “Proof of the ergodic theorem”, ''Proc. Nat. Acad. Sci. USA'', '''17''' (1931), pp. 656–660. {{MR|}} {{ZBL|0003.25602}} {{ZBL|57.1011.02}}
 
|-
 
|-
|valign="top"|{{Ref|KSS}}|| 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 {{MR|0584389}} {{ZBL|0399.28011}}
+
|valign="top"|{{Ref|KSS}}|| 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. {{MR|0584389}} {{ZBL|0399.28011}}
 
|}
 
|}
  
 
====Comments====
 
====Comments====
In non-Soviet literature, the term mean ergodic theorem is used instead of "statistical ergodic theorem" .
+
In non-Soviet literature, the term “mean ergodic theorem” is used instead of “statistical ergodic theorem”.
  
A comprehensive overview of ergodic theorems is in {{Cite|K}}. Many books on ergodic theory contain full proofs of (one or more) ergodic theorems; see e.g. {{Cite|P}}.
+
A comprehensive overview of ergodic theorems is found in {{Cite|K}}. Many books on ergodic theory contain full proofs of (one or more) ergodic theorems; see e.g. {{Cite|P}}.
  
 
====References====
 
====References====
 
{|
 
{|
|valign="top"|{{Ref|K}}|| U. Krengel, "Ergodic theorems" , de Gruyter (1985) {{MR|0797411}} {{ZBL|0575.28009}}
+
|valign="top"|{{Ref|K}}|| U. Krengel, “Ergodic theorems”, de Gruyter (1985). {{MR|0797411}} {{ZBL|0575.28009}}
 
|-
 
|-
|valign="top"|{{Ref|P}}|| K. Peterson, "Ergodic theory" , Cambridge Univ. Press (1983) {{MR|0833286}} {{ZBL|0507.28010}}
+
|valign="top"|{{Ref|P}}|| K. Peterson, “Ergodic theory”, Cambridge Univ. Press (1983). {{MR|0833286}} {{ZBL|0507.28010}}
 
|}
 
|}

Latest revision as of 05:51, 29 November 2016

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.)

References

[B] G.D. Birkhoff, “Proof of the ergodic theorem”, Proc. Nat. Acad. Sci. USA, 17 (1931), pp. 656–660. Zbl 0003.25602 Zbl 57.1011.02
[KSS] 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 Zbl 0399.28011

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

References

[K] U. Krengel, “Ergodic theorems”, de Gruyter (1985). MR0797411 Zbl 0575.28009
[P] K. Peterson, “Ergodic theory”, Cambridge Univ. Press (1983). MR0833286 Zbl 0507.28010
How to Cite This Entry:
Birkhoff ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birkhoff_ergodic_theorem&oldid=39839
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article