View source for Birkhoff ergodic theorem

{{MSC|37A30|37A05,37A10}}

[[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

<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>

(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

<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 [[#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====
<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====
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]]].

====References====
<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) {{MR|0833286}} {{ZBL|0507.28010}} </TD></TR></table>