Birkhoff ergodic theorem
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 . 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 ). 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.)
|||G.D. Birkhoff, "Proof of the ergodic theorem" Proc. Nat. Acad. Sci. USA , 17 (1931) pp. 656–660|
|||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|
In non-Soviet literature, the term mean ergodic theorem is used instead of "statistical ergodic theorem" .
|[a1]||U. Krengel, "Ergodic theorems" , de Gruyter (1985)|
|[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=12514