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 [[#References|[1]]]. 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 [[#References|[2]]]). 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 | + | Birkhoff's theorem was stated and proved by G.D. Birkhoff [[#References|[1]]]. 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 [[#References|[2]]]). 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==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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}} </TD></TR></table> |
====Comments==== | ====Comments==== | ||
− | In non-Soviet literature, the term mean ergodic theorem is used instead of | + | In non-Soviet literature, the term mean ergodic theorem is used instead of "statistical ergodic theorem" . |
A comprehensive overview of ergodic theorems is in [[#References|[a1]]]. Many books on ergodic theory contain full proofs of (one or more) ergodic theorems; see e.g. [[#References|[a2]]]. | A comprehensive overview of ergodic theorems is in [[#References|[a1]]]. Many books on ergodic theory contain full proofs of (one or more) ergodic theorems; see e.g. [[#References|[a2]]]. | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> U. Krengel, "Ergodic theorems" , de Gruyter (1985) {{MR|0797411}} {{ZBL|0575.28009}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Peterson, "Ergodic theory" , Cambridge Univ. Press (1983)</TD></TR></table> |
Revision as of 10:29, 27 March 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 [1]. 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 [2]). 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
[1] | 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 |
[2] | 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 [a1]. Many books on ergodic theory contain full proofs of (one or more) ergodic theorems; see e.g. [a2].
References
[a1] | U. Krengel, "Ergodic theorems" , de Gruyter (1985) MR0797411 Zbl 0575.28009 |
[a2] | K. Peterson, "Ergodic theory" , Cambridge Univ. Press (1983) |
Birkhoff ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birkhoff_ergodic_theorem&oldid=23582