Difference between revisions of "Birkhoff ergodic theorem"
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exists almost everywhere, with the same properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016530/b01653017.png" />. | 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 | + | 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|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 | + | 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}}. |
====References==== | ====References==== | ||
| − | + | {| | |
| − | + | |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}} | ||
| + | |} | ||
Revision as of 20:05, 10 May 2012
2020 Mathematics Subject Classification: Primary: 37A30 Secondary: 37A0537A10 [MSN][ZBL]
One of the most important theorems in ergodic theory. For an endomorphism 
of a space 
with a 
-finite measure 
Birkhoff's ergodic theorem states that for any function 
the limit
 |
(the time average or the average along a trajectory) exists almost everywhere (for almost all 
). Moreover, 
, and if 
, then
 |
For a measurable flow 
in the space 
with a 
-finite measure 
Birkhoff's ergodic theorem states that for any function 
the limit
 |
exists almost everywhere, with the same properties of 
.
Birkhoff's theorem was stated and proved by G.D. Birkhoff [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); 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 theorempointwise 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 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 |
Birkhoff ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birkhoff_ergodic_theorem&oldid=24713