遍历理论 - Wikiwand

遍歷的定義

考慮適定的函數f的時間平均。這定義為從某個初始點x開始的時間間隔T的取值的平均。

{\hat {f}}(x)=\lim _{n\rightarrow \infty }\;{\frac {1}{n}}\sum _{k=0}^{n-1}f\left(T^{k}x\right)

再考慮f的空間平均和相位平均，定義為

{\bar {f}}=\int f\,d\mu

其中μ是概率空間的測度。

一般來說，時間平均和空間平均可能不同。但是若變換是遍歷的，而該測度不變，則時間均值和空間均值幾乎處處相等。這就是著名的遍歷定理，其抽象形式由喬治·戴維·伯克霍夫給出。平均分佈定理是遍歷定理的一個特殊情況，專門處理單位間隔上的概率分佈。

歷史參考

G. D. Birkhoff, Proof of the ergodic theorem, (1931), Proceedings of the National Academy of Sciences USA, 17 pp 656-660.
E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, (1939) Leipzig Ber. Verhandl. Sächs. Akad. Wiss. 91, p.261-304.
S. V. Fomin and I. M. Gelfand, Geodesic flows on manifolds of constant negative curvature, (1952) Uspehi Mat. Nauk 7 no. 1. p. 118-137.
F. I. Mautner, Geodesic flows on symmetric Riemann spaces, (1957) Ann. of Math. 65 p. 416-431.
C. C. Moore, Ergodicity of flows on homogeneous spaces, (1966) Amer. J. Math. 88, p.154-178.

現代參考

Vladimir Igorevich Arnol'd and André Avez, Ergodic Problems of Classical Mechanics. New York: W.A. Benjamin. 1968.
Leo Breiman, Probability. Original edition published by Addison-Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. ISBN 978-0-89871-296-4. (See Chapter 6.)
Peter Walters, An introduction to ergodic theory, Springer, New York, 1982, ISBN 978-0-387-95152-2.
Tim Bedford, Michael Keane and Caroline Series, eds.. Ergodic theory, symbolic dynamics and hyperbolic spaces. Oxford University Press. 1991. ISBN 978-0-19-853390-0. (A survey of topics in ergodic theory; with exercises.)
Joseph M. Rosenblatt and Máté Weirdl, Pointwise ergodic theorems via harmonic analysis, (1993) appearing in Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference, (1995) Karl E. Petersen and Ibrahim A. Salama, eds., Cambridge University Press, Cambridge, ISBN 978-0-521-45999-0. (An extensive survey of the ergodic properties of generalizations of the equidistribution theorem of shift maps on the unit interval. Focuses on methods developed by Bourgain.)